Is the ai in j'ai and finirai (I am assuming they are the same) pronounced exactly like the -er in infinitives: are manger and mangeai pronounced exactly the same?
Both are the same
close-mid front unrounded vowel (
[e]). Continuing my own comparison of -ai and -ais endings using the verb grimper, notice that grimper is listed as having the same pronunciation as grimpé. (Note that they use SAMPA symbols, not IPA.)
Please see the rest of my answer for an audio example of the pronunciation -ai. The third segment of the dictation also has various samples of the pronunciation of the -er ending (for example, Ils se sont entraînés à en déjouer les pièges.)
That said, it seems not everybody uses this pronunciation of -ai so the people who pronounce it like -ais1 will indeed make a difference between -ai and -er (on which there is probably little discussion)
1. They're wrong :-)
No, there's a slight difference in pronunciation between the two forms (although some people in france can't pronounce it differently) :
mangerai (will eat) is pronounced [ɛ] while manger (to eat) is pronounced [e].