My old MBPs motherboard has a faulty HDD connector, are there any other places on the motherboard I could connect storage to? Right now I'm making a persistent Linux usb stick. I don't like the idea having the USB stick plugged in all the time and possibly losing my work if it gets jostled. Is there a work around that will allow me to place a USB stick inside my macbook where the HDD would go?

Or, could the OS X recovery partition be used as a persistent OS?
No - the HDD interface is SATA/AHCI and not USB. I suppose if you were really skilled, you could somehow solder on to the USB inside or re-route one of the connectors / ports.

Have you looked at the extremely tiny USB sticks that barely take much space outside the connector?

Something like SanDisk Cruzer Fit or similar low profile model: https://www.amazon.com/SanDisk-Cruzer-Low-Profile-Drive-SDCZ33-016G-B35/dp/B005FYNSZA