Sounds like you've already chosen a path, but another good option would be the Sansamp GT2. There are kits, schematics, and even used pedals strewn about the web. They have a multitude of great sounds, are relatively low priced, and run from a single 9V source. I always carry one as a spare/emergency rig.

Google says 19 gauge or better. I usually just hack the end off an IEC cable since I have them laying all over the warehouse. 1.8 amps isn't a ton of current, so most any power cable within reason should be able to handle it.

FWIW: Switchcraft makes isolation washers for their jacks.

https://www.tubedepot.com/products/isolation-washers-for-1-4-input-jack-flat-shoulder?srsltid=AfmBOopLRJix5gkjMcz9KdZDMnK0ud8XFPIuZlIIjxTb4ncYBa4kMdX6

That trace looks broken. You will need to scrape off some of the coating and solder across the break. If not completely broken, I'd do it anyway to add stability to the cap leg.

You'll need to resolder that cap leg and maybe jumper the trace to make the connection.  Hard to say for sure without seeing a pic.

FWIW: Unless board mounted and covered up, there is usually an accessible hole in the center of a 9 pin socket. You can use a chop stick or something of the sort to help push the tube out of the socket from the bottom side.

That's not corrosion. It's flux. Not a problem. If you turn the master all the way down (as Juan suggested) is the noise still present? Also, grab the top of those caps and give them a SLIGHT wiggle. That might tell you if any are loose.

That sounds like poor power supply filtering to me. I'd check caps C30, C31, C26, & C27. First, make sure they are properly soldered to the circuit board. It's quite possible you just have a cracked solder joint on one of the caps.