[K7Sparky]

For spiders and basket edges I use toluol and xylene sparingly and plenty of wait time to soften the adhesive. Acetone / lacquer thinner evaporates fast, but works fine to better depending on temperature and the phase of the moon. Sparingly and several times.

I made lots tools out of broken hacksaw and saber blades. Think weird chisels. Some are flat end, some are rounded. some ground offset and use carefully as needed. I grind at around 10 to 15 degrees (swag) and SHARP. Once you get under an edge of the spider slowly moving and rocking the blades lift it. The residual adhesive rubs off without too much trouble.

When cleaning the frame some blue tape over the voice coil gap helps keep the crud out. Standard speaker repair use of tape and compressed air to clean the voice coil gap. Use a good BRIGHT light and plenty of magnification to make SURE the gap is clean. Once is enough take is slow.

Find a couple of old junk worthless speakers to practice on before you attack one you care about. Like anything there is a learning curve littered with destruction.

Take your time with the voice coil cover (Dust cover) in the center. May just want to sacrifice it and buy a modern or make a new felt one. Solvent may loosen the VC windings - no advice.

The warning on shimming (centering) the voice coil is 100% both for spider and cone edge during gluing. Some like shims at 120deg others at 90. I like 90, but can't give you any reason for it. There are special glues for each and they are different. To me it looks and smells like the spider glue is contact cement. The cone is white like a tacky cement from Hobby Lobby, but I admit I buy mine from speaker repair part suppliers.

I like Jamie's idea of a bit of heat. Gun smith screwdriver sets have lots of special ground blades to fit all sorts of special slots if you don't roll your own.