Hello! Is it allowed to beat the game multiple times in "New game +" mode in order to fully upgrade all my weapons, collect green herbs, etc, and then speedrun the game in the same game file?

I think this should be allowed, because if it's not, people are forced to perfectly beat the game in their first run in order to increase their chances in New Game + in their second run.

yes, you are allowed to set up your inventory anyway you want and use which ever weapons/items you want for ng+

Yes you can. You can start a NG+ run from whichever round you feel like it. It doesn't necessarily has to be Round 2.