I've spent countless hours working after hours and sometimes during the weekend to show how I'm engaged with company purpose.
What looks like engagement to you looks like inefficiency to your managers. You are supposed to finish your work in 40 h/week, instead of 60(?) h/week.
During last month he has been complaining a lot about my behaviour when addressing comments in Pull Requests. He said that I'm not taking my time to evaluate each comment and I'm just rushing through comments trying to close my Pull Request as soon as possible.
Then it is possible that you do not a thorough job when analyzing whatever you have to analyze. Doing things fast and doing things good are not the same thing. If your company expects quality before speed, then do not reverse the priorities just because you want to show off.
Should I be worried?
If you actually improve, then no need to worry. At least, not much.
Am I gonna be fired?
Only if you do not show improvement.
How can I manage this situation?
Your manager already did the work for you. You just need to comply. You are actually very lucky. Most of the managers will only tell you about what you did wrong during yearly evaluation, as a pretext to not give you a raise. You manager talks to you when needed, AND you got a raise. It is unreasonable to expect much more.
One thing you can do additionally, is to initiate discussions with your manager, when you are not sure that you understood something correctly. Just be careful not to overdo it. Too many questions will raise another flag in your manager's mind.
My manager does not seem to be a flexible person.
Well, according to your description, your manager is actually a good manager. It seems that it is you who are not flexible: "I've tried to slow down a little bit, but it seems to be enough." (probably "doesn't seem to be enough")
As long as your manager guides you, let yourself guided. I wish I had such managers in my life. I did not, unfortunately.