OP:
Are you making any progress using the USB installer?
If not, and if the internal drive is a platter-based hard drive, I'd like to make a suggestion:
Buy a modest-sized SSD, install that, and install the OS onto it.
If you've just been running with an HDD, an SSD will make a 2010 MBP "feel like a new machine". (It did on MY 2010 MBP).
You don't need a large one -- 256gb or 500gb. I'd suggest a Crucial SSD.
The drive swap is VERY easy.
All you need is a Phillips #00 driver and a TORX T-6 driver.
Go to ifixit.com to see what's involved. Again VERY easy.
While it's open, check the condition of the SATA drive ribbon cable.
Do you see any "wear" where it comes into contact with the back cover?
Sometimes a piece of tape to "protect it" can work here.