Just my opinion but,
I would disassemble and move your dad's machine to a solid surface preferably concrete. Wash , stone and reassemble the clean (notice I didn't say painted) machine parts bottom up.
It sounds like alot of work but a 12 inch lathe is a perfect first time project.
During the assembly you will learn the function and fit of every part.
Grasshopper, you will know your lathe inside and out and when it doesn't function the way you want you will understand what needs to be done.
You can drive a cadillac or a yugo but it's STILL YOU DRIVING.
A nicer more commercial duty machine (cadillac) would be sweet but in the end
Spending some money on grandpa's machine and working on it would be great for both of you and will do most of what you can do on a cadillac.
It shows grandpa financial wisdom, he will like that.
As far as issues on his old machine do what you can on the reassembly and make notes on what you need for repair in the future, but get it right back together .
Another nice thing about grandpa's machine is its simplicity and availability of cheap parts, lots of documentation...
.