Does religion change?

Yes.

If you read the Jewish Bible (the Tanakh) from start to end you will see an evolution of thought. In the earlier books God is spoken of as the God of the Hebrews and reference is made to other gods of other people. YHVH is just One God of many but the only true God of the Hebrews.

Then as you get further you will see the idea evolve that YHVH is the One True God of everyone and all the other gods are false gods. Further still, it is clearly stated that YHVH is the Only God, period.

In more recent times, in the Middle Ages, all three of the Abrahamic faiths agreed, owning slaves was OK, charging interest was not. But as time went on and as the Jews and Christians realized the value of the individual, slavery with them disappeared and today they both agree that slavery is wrong. And at the same time, they both realized that you can not get people to lend money without at least a little benefit in doing it. So today they both allow the charging of a reasonable amount of interest. Both faiths evolved in the past 1000 years. Contrasting that with Islam where slavery was only banned (in Saudi Arabia) in the 1960s when the USA leaned on them in a very forceful manner. And where Islam still does not allow the charging of interest but does allow the lender to be a silent partner of the person who got the loan. (And never being free of the lender, which in itself is a form of slavery as you can never get rid of that silent partner and have to keep paying him his share forever.)

What I would like to see with all religions is a movement to leaving everyone else alone. If someone is not of your faith, you have no right to bother them about it. No one running around claiming their faith is the one true and correct faith. In this, the Muslims and Christians need to work on it.

Some Religions do change. Not all change is bad. When a change occurs, ideally, the traditions etc. of the older version of that Religion are not lost or destroyed.Religions, are not cemented to nor committed to changes determined by what th Age or the World dictates. The Apostles celebrated the Resurrection and also attended the Temple services, at least until they went to other countries to evangelise, There the expression of their religious practises had to fit in with circumstances.

Those Apostles and Disciples who remained in Jerusalem after the Ascension of the Lord, probably practised Judaic customs and rites until the Council of Jamnia excommunicated the followers of the Christ. So, I think that the fundamental roots and even religious knowledge and expressions will remain and change can be for the good but not if the basics are lost, thwarted or perverted.

Religions change as society and culture does, and the holy books will change a little as they get re-translated. So yes, though religious people probably don't like it, their religion does change.

I don't know about answering the second question, but I think people would believe the newer versions are correct (save for fundamentalists and literalists).

Sure they change. How many Christians still stone disobedient children and refuse to eat shellfish, as commanded in their Bible?

I also do not believe that bomb vests had yet been invented circa 600AD, when Islam came about.