No picture is usually the HDMI port
Rule out the easy things first: a different cable, a different input on the television, and a different television if you can. If none of that brings a picture back, the port on the console is the likely cause.
It fails because it is a physical connector that gets levered every time the cable is moved. The port is soldered directly to the board, so the pins can lift away from their pads.
Stop plugging the cable in
If you can see bent pins inside the port, every further attempt bends them more and risks tearing the pads off the board. That turns a routine port replacement into a much larger repair, and occasionally into an uneconomic one.
Bring it in as it is. There is nothing you can do at home that improves the situation.
What the repair involves
The old port comes off with hot air, the pads are cleaned and inspected under a microscope, and the new one is aligned and reflowed. It is microsoldering rather than a parts swap, which is why it takes a day or two rather than an hour.
It costs a fraction of a replacement console, and everything on your drive stays where it is.
Shutting down mid-game is heat
A console that gets loud and then turns itself off is protecting itself. Dust packs into the heatsink fins and the thermal paste between the chip and the cooler dries out, and the machine can no longer move heat away fast enough.
This is the most common console job we do and it is a same-day one. The drop in fan noise afterwards is the part people comment on.
Controller drift is a worn part, not a setting
A stick that moves on its own has a worn potentiometer inside the thumbstick module. Contact cleaner can settle it for a few weeks, but once the track is worn it comes back.
Replacing the module fixes it properly and costs a lot less than a new controller.