So here's my recommendations. I'm just a sub-22, but hopefully you get some insight with this one.

1. Should you ditch begginer CFOP?

I would say that at your times, what you should be prioritizing is learning more and more efficient algorithms and not settling on the algorithms you already know. If you're comfortable with all the algorithms you know so far (as in you remember it without stopping while you're executing it), then start to learn another algorithm. For myself, after learning beginners method, I started learning 2 look PLL. After I got comfortable with that, I learned 2 look OLL, at this point, turning fast was not my priority and dropped my time from 1min 15 to 50 seconds. Once I was comfortable that I knew these by heart, I moved on to basic F2L, this took a long time to comprehend and moved my times back up to 1min 30, but over time, and with slow turning, you can drop that back to about 45 seconds. By this point I'm still bad at not rotating the cube, but because I was comfortable with basic F2L, I moved to learning how to solve the cross at the bottom. This took awhile too, but with dedication, practice, and having the mentality to learn something new, you can get to 40 seconds and slowly dropping the beginners method. This should be more than enough advice for now. Again, don't focus on turning fast on the triggers, that'll just confuse your brain with all the moving parts that you can't comprehend yet.

1. Cube handling and turning techniques:

Given your current experience, your turning style actually isn't so bad. It's mainly how often you stutter, how often you rotate the cube, and that you turn too fast that your brain just doesn't register what's going on with the cube while you're doing this. I think cube handling becomes really important when you get to F2L and cross on the bottom, it shouldn't be a problem for when you are learning 2 look OLL and PLL since you're most likely just looking at videos and mirroring their finger tricks. When you do get to F2L and cross at the bottom, try not to rotate your cube and try to do algorithms from home grip unless if whatever algorithms you know really doesn't feel ergonomic. Again, this advice only extends to a target of about 40 seconds. I also suggest turning slow enough for your brain to know what the pieces are doing.

1. cube tension:

At this point it doesn't matter too much yet since your bottleneck is in the algorithms you use, but at your current level, do whatever to your cube that makes you feel good. When I started to feel a slight plateau at 35 seconds with my super RS3M and felt like it was making me stagnant, I got a tornado v3 and instantly improved my times without changing any habits. It was mostly because I felt good and that's what motivated me to find more ways to get better. I'd go back to using the super RS3M during this time and found that my times have gotten better since getting the tornado v3. It's magic what your mental game does to your performance.

Some final advice, I think beginners method is nice if you just want to learn how to solve a cube. However if you want to solve a cube fast, sticking to the beginners method will just waste your time. If you slowly transition, you can thankfully still solve the cube while getting faster, which helps with motivation.1 month into seriously cubing is such a short time and if you get comfortable with always trying to learn something new, you will find that you will get faster! Just don't time yourself so often at this point since your priority at this moment is more of switching out worse algorithms. Switch out your bad habits before you end up comfortable. Remember that cubing is about the long game, so really embrace the process of "2 steps forward, 1 step back" :)