This is a little bit of a tricky question to answer definitively. There are Right and Wrong approaches, there are multiple opinions on what is the best method. I am going to try and approach this somewhat neutrally with my thoughts based of your statements.
1. What's a better timed game to learn from? I usually play 10 minute games, but I've considered switching to 30 minute games until I get better.
There are two camps here. Those who say faster games as it helps with pattern recognition. There are those who say longer time controls so you can train analysis and depth. Each camp is right in their own way. Pattern recognition is important, bullet and blitz time controls are quite good at developing that. Rapid games are good analysis. Daily Games are extremely analytical...possibly too much so.
Based of your response that your making a lot of the same mistakes. I think pattern recognition might be a bit of an issue for you at this time. I would actually be suggesting you look into doing Tactics puzzles for 10 to 15 minutes before you play a game with a longer time control say 10 to 15 minutes again. Playing bullet games which rely heavily on pattern recognition when you haven't developed that skill sufficiently is just not a productive endeavour.
2. I keep making a lot of the same mistakes, and I'll readily admit I likely have very, very poor strategy. Any tips on improving there?
If you know what the mistakes you are making , such as hanging pieces, you can use the chess.com puzzle functionality and if you click on customise then you will see themes. Select the themes for which you are commonly making mistakes on and set the ratings to be between your current rating and maybe 500 above. Spend time on the positions. Its ok to spend an hour on a puzzle. You are far better to spend time analysing the position getting it right than spending 30 seconds and going "I don't know..." and just moving to progress.
Strategy is very different to train than Tactics and is really about the analysis of the "long" game. Reviewing the board , developing a plan and executing it. The following article provides a reasonable overview. Main thing to remember though is to have a plan...but not to get to attached to that plan.
3. Is there a way (or a website or something) that I can put in the moves after a game (the PGN) and find out why a move was marked as it was (good, excellent, blunder, etc.)?
As you are using Chess.com I would actually suggest using the in built analysis tool.
You will notice there is notation after the best suggested move. This is how the engine would interpret the position to be developed if played "correctly".
You can even click on the magnify glass to copy the line to the overall game annotation as a variation
In regards to seeing why the move is better or a blunder it is a matter of reading or playing out the variation. Adding it to the analysis allows for it to actually be played out as a variation.
I would suggest reading the following article from chess.com though as its quite comprehensive.
If you are willing to make an investment in your learning you could look into applications like DecodeChess which is a reasonably good platform and on the extreme end Chessbase which is a desktop application (extremely pricy though).
I personally prescribe to the longer time intervals for my students as I believe that the process of Analysis significantly improves player skill. Pattern recognition is also extremely important which is why there is also the Tactical Analysis (Puzzles)
This is a very simplified stripped back approach but generally all good training should be balanced between effort,rest and available time.
Rapid 15 Minute games - 5 to 10 games / week
- Analysis of each game
Blitz 5 Minute games - 0 to 5 games / week
- Analysis of each game
Tactics Puzzles - 10 puzzles a day (exc Rest Days).
One to Two days of the week you don't touch anything chess related.
This sort of format I find generally works well for players between beginner to 1500 range. Focus on maybe learning 3 to 4 openings, the most popular ones focus on the first 4 to 5 moves. This will help set you up.
There are commercial sites like AimChess.com as well provide training programs and generally work on a similar principal. Which I also generally recommend.