(LDS view) Do animals have a possibility to eternal life and is this dependent on Jesus atonoment applied to them? This is my other question that was inspired by what a quotation in this other question says about pre-existence prior to conception.

The highest rated answer quotes from The Encyclopedia of Mormonism, on the page Pre-Existence (Pre-Earthly Existence) that says:


  It is Latter-day Saint doctrine that living things existed as
  individual spirit beings and possessed varying degrees of intelligence
  in an active, conscious spirit state before mortal birth and that the
  spirit continues to live and function in the mortal body. [emphasis added]


While reading the previous quote, I noticed a link to this page titled "Animals" which states that:


  Latter-day Saints believe that animals, like humans, have spirits, in
  the form of their bodies (D&amp;C 77:2). Like humans and plants,
  animals were created first as spirits in heaven and then physically on
  the earth (Moses 3:5). Mortal and subject to death, animals
  will be saved through the Atonement of Christ (TPJS, pp. 291-92). [emphasis added]


The last sentence refers to pages 291-292 of "TPJS" which is an acronym for the the book, "Teachings of the Prophet Joseph Smith." In the aforementioned pages, the book says the following: 


  I suppose John saw beings there of a thousand forms, that had been
  saved from ten thousand times ten thousand6 earths like this,—strange
  beasts of which we have no conception: all might be seen in heaven.
  The grand secret was to show John what there was in heaven. John
  learned that God glorified Himself by saving all that His hands had
  made, whether beasts, fowls, fishes or men; and He will glorify
  Himself with them.
  
  Says one, “I cannot believe in the salvation of beasts.” Any man who
  would tell you that this could not be, would tell you that the
  revelations are not true. John heard the words of the beasts giving
  glory to God, and understood them. God who made the beasts could
  understand every language spoken by them. The four beasts were four of
  the most noble animals that had filled the measure of their creation,
  and had been saved from other worlds, because they were perfect: they
  were like angels in their sphere. We are not told where they came
  from, and I do not know; but they were seen and heard by John praising
  and glorifying God.


This same thing is referenced in this August, 1972 Ensign article "The Gospel and Animals" which reads:


  While revising the Bible, the Prophet desired further understanding
  concerning the four beasts mentioned in Revelation 4:6 [Rev. 4:6].
  Section 77 of the Doctrine and Covenants contained the response from
  the Lord. The answer revealed that “heaven, the paradise of God,”
  contained beasts, creeping things, and fowls of the air, and “every
  other creature which God has created.” (D&amp;C 77:2.)




I have a couple of questions I would like to ask about this:

1. Is it the LDS Church teaching that the value of Jesus' blood is also
    applied to save animals?

2. According to the LDS Church, do all animals live forever or is this
    dependent on them "accepting Jesus" or something like that?
Not much is written about animals going to heaven. In answer the OP quotes from EoM Animals and here is the whole first paragraph with answers (#1 bold, #2 italics):


  Latter-day Saints believe that animals, like humans, have spirits, in the form of their bodies (D&amp;C 77:2). Like humans and plants, animals were created first as spirits in heaven and then physically on the earth (Moses 3:5). Mortal and subject to death, animals will be saved through the Atonement of Christ (TPJS, pp. 291-92). Humans and animals will eventually live in peace on this earth (Isa. 11:6-9; 2 Ne. 30:12-15; D&amp;C 101:24-26). The Prophet Joseph Smith taught that animals will be found in heaven, in myriad forms, from myriad worlds, enjoying eternal felicity, and praising God in languages God understands (TPJS, pp. 291-92).


In summary, yes animals are saved through Jesus Christ's blood and animals will live forever (independent of their belief).