We also have scriptures such as D&C 1:38, D&C 21:4, etc.. that appear to say that the prophets are "infallible". Is this correct? Are they infallible?

There are several quotes from the prophets and scholars which have denied infallibility. A search on LDS Library or other LDS databases will quickly show this. The above mentioned scriptures also don't show infallibility. D&C 1:38 says that the Lord's word will all be fulfilled by either the Lord's voice or the voice of the prophets. In both cases, the word of the Lord is what is advanced, not the word of men. The prophets are simply to speak what the Lord has told them. This is also the case with D&C 21:4 when it says that we are to obey the words of the prophet "as he receiveth them". In other words, it is the word of the Lord which we are to obey, not the arm flesh.

Getting back to President Woodruff's comment about the "President of this Church, (not being able) to lead you astray", what does that mean? Does it mean that the prophets will never make doctrinal or predictive claims that are false? Does it mean that they will never contradict scripture? Or does it simply mean that any errors made by the prophets will be minor and not derail the Lord's work thereby leading the Church into apostasy?

Joseph Smith, Harold B. Lee, Joseph Fielding Smith and others have stated that it is not impossible for the prophets to teach things contrary to the scriptures. Consider the following quote -

"If I ever say anything which is contrary to the scriptures, then the scriptures prevail."
(Mark E. Petersen, Adam: Who Is He? [Salt Lake City: Deseret Book Co., 1976], 15.), (Church News, Aug. 23, 1975).

Obviously he felt that it wasn't impossible for the prophet to say something contrary to scripture. Harold B. Lee said -
"If anyone, regardless of his position in the Church (emphasis added), were to advance a doctrine that is not substantiated by the standard Church works, meaning the Bible, the Book of Mormon, the Doctrine and Covenants, and the Pearl of Great Price, you may know that his statement is merely his private opinion. The only one authorized to bring forth any new doctrine is the President of the Church, who, when he does, will declare it as revelation from God, and it will be so accepted by the Council of the Twelve and sustained by the body of the Church. And if any man speak a doctrine which contradicts what is in the standard Church works, you may know by that same token that it is false and you are not bound to accept it as truth."
(Harold B. Lee, The Teachings of Harold B. Lee, edited by Clyde J. Williams [Salt Lake City: Bookcraft, 1996], 544.)

This sentiment is echoed here by FAIR, the premier LDS organization / website dealing with LDS apologetics.

President Charles W. Penrose of the First Presidency, once wrote: "We do not believe in the infallibility of man. When God reveals anything it is truth, and truth is infallible. No President has claimed infallibility."
(Editor's Table, [Improvement Era, September 1912]: 1045.)

Prophets have also taught things that turned out to be factually incorrect. Brigham Young taught that men live on the moon and sun - "Who can tell us of the inhabitants of this little planet that shines of an evening, called the moon? When we view its face we may see what is termed "the man in the moon," and what some philosophers declare are the shadows of mountains. But these sayings are very vague, and amount to nothing; and when you inquire about the inhabitants of that sphere you find that the most learned are as ignorant in regard to them as the most ignorant of their fellows. So it is with regard to the inhabitants of the sun. Do you think it is inhabited? I rather think it is. Do you think there is any life there? No question of it; it was not made in vain."
(Journal of Discources, vol. 13:p.271).

Joseph Fielding Smith said that "it is doubtful that man will ever be permitted to make any instrument or ship to travel through space and visit the moon or any distant planet."
(Joseph Fielding Smith, Answers to Gospel Questions, 5 vols. [Salt Lake City: Deseret Book Co., 1957-1966], Vol. 2, Ch. 43, Pg. 190.).

Other examples of the prophets making mistakes include the Church buying forged documents from Mark Hofmann, the failed 18-month mission term for Elders which resulted in plummeting baptisms and was therefore quickly rescinded (see below), and the ban on flights for missionaries and General Authorities over the Y2K time period though no problems occurred anywhere in the world. These, along with others, show that not everything from Salt Lake is inspired, but instead comes from the personal opinions, limited human reasoning and the knowledge, experience and logic of Church leaders and/or their advisors. The scriptures also indicates that prophets can be mislead by others into making bad decisions. Two examples are found in Joshua 9 and 1 Kings 13 where prophets were deceived. Even Joseph Smith made judgments in error and was rebuked by God for them.

On January 5th, 1982, the First Presidency issued an official statement saying that oral sex was an "unholy and impure practice" and must be avoided by LDS. (See Page 1 and Page2 if you'd like to read the letter. Left click on the photos to enlarge them). It is generally believed that the original statement was prompted by women complaining to their bishops about their husbands pressuring them to provide them with oral sex. This policy statement was withdrawn nine months later and those LDS getting married today and married converts are not informed of this opinion. The current policy now has leaders refraining from asking members about their sex lives. If the Lord really felt that oral sex was an "unholy and impure practice", don't you think that people would and should be warned against it so as to not inadvertantly violate their temple covenants by engaging in it? Obviously it is not an "unholy and impure practice" and the First Presidency, by rescinding their letter, basically admitted their mistake in trying to impose their subjective opinions on the Church.

In April of that same year, 1982, the Prophet changed the length of service for full-time single elders lowered from 24 months to 18 months. The justification for the change: "It is anticipated that this shortened term will make it possible for many to go who cannot go under present financial circumstances. This will extend the opportunity for missionary service to an enlarged body of our young men."

Year	Missionaries	Convert Baptisms	Change from 1981
1980	29,953	211,000	N/A
1981	29,700	224,000	N/A
1982	26,300	207,000	-17,000
1983	26,565	189,419	-34,581
1984	27,655	192,983	-31,017
1985	29,265	197,640	-26,360
1986	31,803	216,210	-7,790
1987	34,750	227,284	3,284

One problem though - that didn't happen. There was no increase in the number of young men (or young women) serving missions as was anticipated. This static supply and the shortened term of service resulted in fewer missionaries out in the field at any given time since the total numbers of missionaries were only those who had left on their missions in the prior 18 months rather than those who left in the prior 24 months. This resulted in fewer missionaries in every mission and mission presidents were forced to assign fewer missionaries to any given area. Some areas lost the only pair they had thereby curtailing most missionary efforts in those areas.
Since they were also being sent home earlier than in the past, just as they reached their maximum proficiency in both language and presentation skills, the missionaries in the field were overall less experienced. These two factors - fewer missionaries and missionaries less skilled overall - caused convert baptisms to plummet as this table shows. The numbers in the first three columns are hotlinks to the Church's website so that the reader can verify them. The policy was reversed in January, 1985. It took the Church six years to recover and reach the levels of convert baptisms as were experienced prior to the change. 116,748 baptisms were lost assuming that baptisms would have remained flat from 1981 through 1986. Since convert baptism numbers historically trend upward, it isn't a stretch to believe that 200,000 convert baptisms were lost while trying to catch up to the previous pace.
Do you really believe that God inspired the 1982 change?

Joseph Smith, Joseph Fielding Smith and Harold B. Lee (and perhaps other prophets) all taught that we should ignore the words of any man which contradict scripture. Please visit our essay - "What Do The Prophets Say About Their Words vs. Scripture?" to see their words indicating that scripture supersedes their words and that their words contradicting scripture are to be ignored. Since there have been times when the prophets have said wrong things as outlined above and that they may say some things contrary to the scriptures (See also "Do The Prophets Always Interpret The Scriptures Correctly?" for other examples), we can clearly see that President Woodruff's statement does not mean that the prophets are never wrong. The Brethren have consistantly rejected the idea that they are infallible. They therefore are subject to making mistakes as shown above.

Joseph Smith also taught that it is wrong to follow leaders when they are wrong - "We have heard men who hold the priesthood remark that they would do anything they were told to do by those who preside over them [even] if they knew it was wrong; but such obedience as this is worse than folly to us; it is slavery in the extreme; and the man who would thus willingly degrade himself, should not claim a rank among intelligent beings, until he turns from his folly. A man of God would despise the idea. Others, in the extreme exercise of their almighty authority have taught that such obedience was necessary, and that no matter what the saints were told do by their presidents they should do it without any questions. When Elders of Israel will so far indulge in these extreme notions of obedience as to teach them to the people, it is generally because they have it in their hearts to do wrong themselves."
(Joseph Smith - Millennial Star, Vol 14, Number 38, pages 593-595).

Since the prophets are also mortal men, even they are subject to error. Many believe that they are the Lord's ventriloquist dummy whose mouth only opens when the Lord pulls the strings.

Please visit our essay - "What Do The Prophets Say About Their Words vs. Scripture?" to see their words indicating that scripture supersedes their words and that their words contradicting scripture are to be ignored. See also "Do The Prophets Always Interpret The Scriptures Correctly?" for other examples of where the prophets have misinterpreted scripture.

The bottom line is that the prophets have been wrong and asking the Church to follow them when they are wrong, is also wrong.