MacBook Pro Mid 2010 13" freezes/restarts when applying pressure on specific part on the back
I used to repair these computers, but I'm not up-to-date with service options. My answer assumed that the computer hasn't been repaired before.

tl;dr: I suggest visiting an Apple Store or an Apple Service Provider and asking for a new top case, bottom case, and logic board.

It sounds like you're describing two issues:


The MacBook crashes when you press in a certain area.
The MacBook randomly crashes with the "you need to restart your computer" message.


The second issue sounds like a kernel panic. That might be a software issue, but more often than not, it's a logic board hardware issue in these Mid-2010 MacBook Pros. There used to be a quality program (or replacement program) for the logic board's video issues, but it has ended. So, odds are, you're overdue for a replacement logic board. But eight years is pretty good for a computer that has survived falls and third-party modifications.

However, replacing the logic board might not resolve the first issue. It sounds like there is something shorting out the logic board, which causes the computer to restart. The "something" might be a dent in the top case (the top part of the computer containing the keyboard, power button, and trackpad) or the bottom case (the bottom part screwed into the top case) making contact with the board. I've seen cases like this before where the customer kept coming back for the logic board every time, when it was a dented bottom case all along.

In any case, you definitely do need a new logic board if any part of it is damaged. A service technician might be able to offer a more definitive answer. I don't think it's very expensive to get these computers repaired at an Apple Store (it's at least cheaper than a new computer), and I don't think that Apple Stores care very much about third-party modifications.