[Dave]

Before bidding for a variac or isolation transformer on eBay, visit some electronics swap meets or amateur radio shows. They turn up very frequently and since they are pretty heavy items, you won't have to pay a fortune on shipping (it's also a good place to find old radios, parts & test equipment). Last year I picked up a massive antique variac for $20 at a local hamfest. Look for one that can handle a current load of at least 2-3 amps, which is plenty for most radios, although radios with high tube counts may require something a bit beefier.
An isolation transformer isn't a requirement for working on radios, but it's a lot safer if you use one. Think of them as an inexpesive insurance policy. Look for one that has a 120 volt output and an adequate current supply. There are a few techniques that you should follow to help prevent leathal shocks which are also a good practice even if you do use an isolation transformer. When working on a live radio, keep one hand behind your back at all times, use tools with well insulated handles, don't wear jewelry or a metal watch and if you must touch the radio, use the back side of your finger (if you do happen to get shocked, your muscles will tense up and pull your finger away rather than latching on & causing more harm).
As for the wire sheathing, most folks prefer to use heat shrink. Use the smallest diameter that will still easily slide over the wire. Replacing the wires is a little more tricky, but it's usually fairly painless if you follow these rules... use the same gauge or slightly larger, cut it to the same length and route it exactly the same as the original. It will help greatly to take lots of photographs from every possible angle before you begin any work. That way you can refer back to them to make sure everything gets put back the way it was.
Dave