Were Adam and Eve meant to live forever; had they not disobeyed God?

No, Adam and Eve were not meant to live forever. If they had been, God would not have made the pronouncement in Genesis 3:22-23:

"and now, lest he put forth his hand, and take also from the tree of life, and eat, and live forever"­--Therefore the Lord God sent him forth from the Garden of Eden...

There is much to be considered here beyond just the tree of life and the tree of the knowledge of good and evil. We must first take a look at what happened when God created man in the beginning.

All Scripture is quoted from the King James translation.

Genesis 2:7 And the LORD God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.

Notice the part about becoming a living soul. The soul is an eternal part of man. Jesus himself gave us some insight into this in the book of Matthew.

Matthew 10:28 And fear not them which kill the body, but are not able to kill the soul: but rather fear him which is able to destroy both soul and body in hell.

Here Jesus was referring to the second death, which we learn about in the Revelation.

Revelation 20:14 And death and hell were cast into the lake of fire. This is the second death.

And the soul of man will not be destroyed nor taste death until:

Revelation 20:15 And whosoever was not found written in the book of life was cast into the lake of fire.

So in answer to your question we cannot ascertain which any veracity whether or not God intended them to have eternal physical life. However; we can without fear of contradiction most assuredly assert that God intended the soul to live for eternity. Which it will. All souls are in a state of waiting after physical death until the final judgment. At that time Jesus will judge all of mankind:

Revelation 20:12 and 13 And I saw the dead, small and great, stand before God; and the books were opened: and another book was opened, which is the book of life: and the dead were judged out of those things which were written in the books, according to their works. And the sea gave up the dead which were in it; and death and hell delivered up the dead which were in them: and they were judged every man according to their works.

And since Death has already been cast into the lake of fire, the soul which are relegated to the Lake of fire will never die but suffer for eternity.

To my mind whether or not God intended that Adam and Eve and their posterity live a physical life throughout eternity is of very little importance. The important part to my mind is the eternal destination of my soul.

Those are my thoughts, you must come to your own conclusions.

Yes.

As @Brasshat cites in his answer, Genesis 3:22 shows that to take from the Tree of Life allows one to live forever, and Adam and Eve were expelled from the garden specifically so that they could not eat from the Tree of Life and live eternally.

Messiah reveals in John 17 that to live forever means to know the Father and the Son, in the sense of familiarity.

It follows that had Adam and Eve not sinned, they would have had access to the Tree of Life and would have been able to eat continuously of the tree, and therefore have eternal life. The specificity of the reference to the Tree of Life in Genesis 3:22 suggests that it was God's intention for them to remain in the garden and continue to have access to the Tree of Life unless they sinned. Because they sinned, they were cut off from eternal life, and had no access to the source of life, which is the Tree of Life, and separation from the Source of Life is the death of which God warned them when he told them not to eat of the tree of the knowledge of good and evil.

So, because they would have remained in communion with the Source of Life if they had not sinned, they would have had life indefinitely.

According to Christian tradition, Messiah is the Source of Life. He says:

"I am the true vine, and my Father is the Grower. Every branch in me that is not bearing fruit he takes, and every one bearing fruit he prunes, so that it bears more fruit"

"I am the vine, and you are the branches. The one who dwells in me bears many fruit, but apart from me you can do nothing."

"I am the bread of life. The one who comes to me will not be hungry, and the one who believes in me will not ever thirst."

"I am the bread of life. Your ancestors ate who knows what in the wilderness, and died. This bread, the one that descends from Heaven, whoever eats of it will not die. If anyone eats this bread, he will live into the ages."