Before the fall of Adam, did anything die?

I think it is important to understand that death was created by God himself. However, death was powerless to do any harm. Death was in the garden of Eden but needed 'disobedience' to activate it. Man (Adam & Eve) was placed in a garden where death was. Gen. 2:15 "And the LORD God took the man, and put him into the garden of Eden to dress it and to keep it. 16 And the LORD God commanded the man, saying, Of every tree of the garden thou mayest freely eat: 17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die."
If man had not disobeyed the word of God, death would have remained powerless over man and so to redeem man it is only the Word that has the capacity to do it - Christ. John 1:1-5. Since death means separation from God, it is therefore safe to say Satan and his demons had died before the garden of Eden. Ezekiel 28:15; Rev. 12:7-9. Satan was already dead.

Death entered the world with Adam's sin, so therefore death wasn't present before the fall.

People and animals ate seeds and fruit.

And God said: Behold I have given you every herb bearing seed upon the earth, and all trees that have in themselves seed of their own kind, to be your meat: And to all beasts of the earth, and to every fowl of the air, and to all that move upon the earth, and wherein there is life, that they may have to feed upon. And it was so done. Genesis 1:29-30

Dying plants don't count because things without blood are not considered alive.

For the life of the flesh is in the blood, and I have given it to you upon the altar to make atonement for your souls; for it is the blood that makes atonement for the soul.’

Leviticus 17:11

Isaiah 11:6 talks about a future paradise where animals don't eat each other

The wolf will live with the lamb, the leopard will lie down with the goat, the calf and the lion and the yearling together; and a little child will lead them.

The views of Christians on this matter can basically be divided into two.

A key passage here is Romans 5:12-21, which says:

Therefore, just as sin entered the world through one man, and death through sin, and in this way death came to all people, because all sinned— To be sure, sin was in the world before the law was given, but sin is not charged against anyone’s account where there is no law. Nevertheless, death reigned from the time of Adam to the time of Moses, even over those who did not sin by breaking a command, as did Adam, who is a pattern of the one to come. But the gift is not like the trespass. For if the many died by the trespass of the one man, how much more did God’s grace and the gift that came by the grace of the one man, Jesus Christ, overflow to the many! Nor can the gift of God be compared with the result of one man’s sin: The judgment followed one sin and brought condemnation, but the gift followed many trespasses and brought justification. For if, by the trespass of the one man, death reigned through that one man, how much more will those who receive God’s abundant provision of grace and of the gift of righteousness reign in life through the one man, Jesus Christ! Consequently, just as one trespass resulted in condemnation for all people, so also one righteous act resulted in justification and life for all people. 19 For just as through the disobedience of the one man the many were made sinners, so also through the obedience of the one man the many will be made righteous. The law was brought in so that the trespass might increase. But where sin increased, grace increased all the more, 21 so that, just as sin reigned in death, so also grace might reign through righteousness to bring eternal life through Jesus Christ our Lord.

Young Earth Creationists - before Adam sinned, nothing died

In the literalist view, the interpretation of the passage above is that Death is punishment for sin, and nature as subservient to mankind shares in Adam's fall. Therefore all death, even of animals, is a result of sin, and did not occur until Adam sinned.

This passage is used as one of the chief Biblical arguments that Genesis 1 should be taken literally and not metaphorically. If there were indeed millions of years of evolution before Adam, then these could not have happened without death and this passage would be contradicted.

As you already stated, Adam and Eve were vegetarians, so there was no need for there to be animal death in order for them to eat. (In fact it is not until Noah that humans are allowed to eat animals).

Other Christians - Physical death of animals did in fact happen before the fall

If you do not hold to the Young Earth beliefs then it is clear that death of animals must have occurred before the fall. The interpretation of the Romans 5 passage is that it is about only the death of humans - death for animals is entirely natural and not the result of sin. (The NIV quoted above takes this line in its translation, but not all translations do). Some also take it to mean 'spiritual death' - i.e. that humans were always destined for a physical death, but that people would know that they were destined to pass on after death to be with God, and it would be entirely natural and nothing to be afraid of. After the fall humans would also suffer spiritual death, i.e. eternal separation from God. The Bible does elsewhere use the word 'death' to refer to spiritual death and not physical death.

If you hold that the Fall is metaphorical rather than literal then it isn't an issue.