If I do all the required radiant quests for Under New Management before finishing the Thieves Guild main storyline, do I instantly become Guild Master and get the armor after Killing Mercer Frey, or will that break the game? Also, is it ok to keep quitting assignments when you've finished the required special job in a hold and Delvin and Vex keep sending you there?

Yes and yes.

A tip on the radiant quests is instead of going through all the dialog and having to refuse quests you don't want is to instead let the Delvin or Vex give their final flavor dialog line (".... let me give you the details") and quickly exit out of it before their voice over finishes and save your game. The random location for a particular quest is determined at the exact moment the quest officially starts, if you can save after verbally getting the assignment but before it shows up as a quest in your journal you can play with it to get the location you want, reloading right before that exact moment the choice happens. Obviously this works best if you have the game installed on an SSD but then what doesn't.