1. "Mar/Mara" indeed means bitter, and "Yam" means sea.
And if we check out the first chapter in Exodus (15) where the name Miriam appears, there seems to be a very close connection between the name and this meaning. The chapter is about the drowning of the Egyptians in the sea by god, and later goes on to Moses and his people looking for water in the desert, and when they found water they couldn't drink it, because it was bitter (and then god made it sweet, etc.)
Also as you pointed out correctly, the name "Miriam" as it is written in Hebrew may be read as "Mariam", since there is no vowel after the "M".
So it's seems very clear how this theory came about.
2. The word "Meri" also means "rebellion".
3. As for the meaning of "wished for a child" - I can't say were it came from. Same for "beloved", But as Anneza and this site points out, an Egyptian name seems very likely (Miriam was born in Egypt).