So if mistreating animals is wrong and we are not supposed to be the "authors of misery to any part of creation." (Brigham Young: ID 11:150) then does that mean that we shouldn't eat them?

But note importantly that God's permission given to Noah to eat meat is followed with a warning: "But, the blood of all flesh which I have given you for meat, shall be shed upon the ground, which taketh life thereof and the blood ye shall not eat. And surely, blood shall not be shed, only for meat, to save your lives; and the blood of every beast will I require at your hands." (JST Genesis 9:10-11) This scripture seems to be saying that the only time we should kill animals is if we would starve otherwise.

There are several other scriptures that give us permission to eat animals, especially in the Doctrine and Covenants. But they are followed with a warning to eat it only when necessary. D&C 49 teaches: "And whoso forbiddeth to abstain from meats, that man should not eat the same, is not ordained of God; For, behold, the beasts of the field and the fowls of the air, and that which cometh of the earth, is ordained for the use of man for food and for raiment, and that he might have in abundance. "(D&C 49:15, 18-19) This seems to say that forbidding meat eating is wrong, since God gave permission to eat animals. But note that the very next verse gives a strong warning: "But it is not given that one man should possess that which is above another, wherefore the world lieth in sin. And wo be unto that man that sheddeth blood or that wasteth flesh and hath no need." (D&C 49: 20-21). So God does not support the "forbidding" of eating meat; but he also does not support eating meat or killing animals in any way unless we need to.

D&C 59 also teaches us that "all things which come of the earth, in the season thereof, are made for the benefit and the use of man ..." but the next verse clarifies: "for unto this end were they made to be used, with judgment, not to excess, neither by extortion." (D&C 59: 18-19) The mistreatment of animals on today's factory farms could easily be referred to as 'extortion'.

The famous word of wisdom (D&C 89) also teaches us about the eating of animals. It says:
Yea, flesh also of beasts and of the fowls of the air, I, the Lord, have ordained for the use of man with thanksgiving; nevertheless they are to be used sparingly; And it is pleasing unto me that they should not be used, only in times of winter, or of cold, or famine.
All grain is ordained for the use of man and of beasts, to be the staff of life, not only for man but for the beasts of the field, and the fowls of heaven, and all wild animals that run or creep on the earth; And these hath God made for the use of man only in times of famine and excess of hunger. (D&C 89:12-15.)

Additionally, the word 'sparingly' not only means "in great moderation," but also "in a sparing or saving manner." This word of wisdom may be expressing a concern for animal life as well as for our health. An article about the word of wisdom that appeared in the Times and Seasons during the lifetime of Joseph Smith wrote:
"Let men attend to these instructions; let them use the things ordained of God; let them be sparing of the life of animals...." (Times and Seasons III [June 1, 1842]: 801.)

These statements of scripture are supported by many LDS prophets and apostles, too many to mention here, but I share a few:

The prophet Joseph Smith, wrote: "I exhorted the brethren not to kill a serpent, bird, or an animal of any kind during my journey unless it became necessary in order to preserve ourselves from hunger." (DHC 2:71.) Journals report that he repeatedly made sure that the latter-day pioneers were following this lesson. 
George Q. Cannon (1st counselor in the presidency) wrote on killing animals: "... Am I or my family hungry? If so, of course man is justified in killing animals or birds to satisfy his or his family's hunger. But if he has not any want of meat he "sheddeth blood," and he exposes himself to this wo which the Lord has pronounced." (Juvenile Instructor 34 [Oct 1,1899]: 592.)

The prophet Joseph F. Smith: "I do not believe any man should kill animals or birds unless he needs them for food..." (J148 [May 1913]: 308-309.)

Hyrum M. Smith: "To kill, when not necessary, is a sin akin to murder." (D&C Commentary, p. 286.) 
Heber C. Kimball: 'There is nothing in the spirit of love that will kill or destroy unnecessarily...." (JD 6:128.)
President Loreuzo Snow said he "thought the time was near at hand when the Latter-day Saints should be taught to refrain from meat eating and the shedding of animal blood." (Journal History, November 3, 1897.) 
The prophet Joseph Smith also taught: "Men must become harmless before the brute creation, and when men lose their vicious dispositions and cease to destroy the animal race, the lion and the lamb can dwell together, and the suckling child can play with the serpent in safety." (TPJS; p. 71)

Viewing the practices on modern-day factory farms, the light of Christ plainly teaches that such maltreatment of our fellow beings grieves the Spirit of our Heavenly Father. A conscience-full latter-day saint then should avoid supporting such things. The prophet Brigham Young taught: "If we maltreat our animals, or each other, the spirit within us, our traditions, and the Bible, all agree in declaring it is wrong,"(JD 1:336-337) and further:
"Were it not for the ignorance of the people, the Lord would curse them for such things. " (ID 15:227)

Let us, then as latter-day saints, seek to refrain from supporting anything that harms our fellow beings, and our own health, so that we can be part of bringing forth that beautiful day when: "the enmity of man, and the enmity of beasts, yea, the enmity of all flesh, shall cease from before my face" (D&C 101:24-26) and the lion and the lamb can finally lie down together "And they shall not hurt nor destroy in all my holy mountain." (Isaiah 11:6-9)