Will TVHM/Mayhem be unlocked if I finish the game in easy difficulty?

Solution 1:

So, long answer short: YES.

I can't link any official statement and I even can't talk from own experience, but I am 99.9% sure that you will be able to continue with TVHM after completing the normal game on easy difficulty.

Two things lead me to this: First, you can switch between easy and normal difficulty any time you want. Second, it would be a very bad game design decision to not allow player continue with TVHM (which basically means Levels 35 and above are locked away). Even further it would be a very very bad game design decision to not explicitly mention it to the player while giving the choice to select easy mode. Something along the lines like "If you complete normal mode on easy difficulty, you will not be able to continue with TVHM".