Yes, working on tactics problems helps improve your chess. TT rating is time dependant so blitz may have sped you up. Also, TT rating is how good you are at blitz tactical exercises not how good you are at finding tactics in general.
You get out of practice what you put into practice. In blitz games you have to cut calculation short and can't do any analysis. There's no time to "find a good move, look for a better one" and push the limits of what you know and are able to do. You only have time to play what you know very well. To push your game further you have to challenge your previous ideas.
5/2 is fine until you play in a tournament and realize you have no idea what to do when both players have multiple hours. When I began tournaments after many years of blitz my thinking was very disorganized and I took too much time and got into time trouble. I could find OK moves very fast, but analysis was new to me. Others have the opposite problem and can't slow down. They're ahead on the clock but losing on the board.
Analysis, which doesn't happen in blitz games but all the time in long games, is calculating a candidate move to the end and assessing an evaluation. Then calculating a different candidate move, assessing its evaluation, and finally comparing the two to decide which is better.
Does solving tactics improve chess or does playing chess improve solving tactics?
My TT rating has increased after a few rounds of blitz chess.
Also, another question is, is it really better to play 15 10 or 30 0 chess? And why? People always say that it is better but never say why. I find it okay to play 5 2.