I have experienced the same hesitation that you’re referring to suppose most of us have. What I do is simply protect what might be under the area you’re soldering with a shop towel. Just carefully wrap it around the underside so that if anything does fall, and it may well, it falls onto the sacrificial surface and not onto anything critical that could be damaged.

Just take your time. As for desoldering first, it is not absolutely necessary.

Just go slow and be as careful as you can be. I’m sure you will be successful

That looks like the lug snapped you could still resolder it you don't need to desolder anything first since it's a coil you will need lots of heat this is one area where a 100 watt gun is warranted.

Get sweet. Get some scrap wire, lead solder. Sit at a bench or table and practice. Youtube has lots of short videos to help. Learn to heat a joint and flow the solder.

Can be fixed. Not an easy task. Maybe get a new coil and start over fresh.

LTG : )

The "hack" approach would be to cut off the lug at the wire and solder the wire directly to the remaining lug on the coil.

Soldering is easy peasy. What you need is an iron that gets hot enough. With a hot iron, it can be done in seconds.

Strip off the end of the wire, tin the wire (add solder to the wire), "tin" the lug (or what's left of it), then heat up the lug and the tinned wire at the same time. When both are hot, touch them together and finish it up with a touch of additional heat.

Can de done under the playfield.

This is fairly common, so don't worry too much about it. The flipper coils get a lot of vibration and sometimes things break.

The secret to soldering is adding more solder. (It's actually about the flux in the solder getting to where everything can merge together), so the advice above where you use a towel below where any excess solder can drip is a very good idea.

If you use a soldering GUN, it is essential that the tip of the gun never had been used with acid cored plumber solder. Even though you are using rosin core electronics solder now, prior use of acid core solder is a bad idea for electronics.

Lots of good advice here, don't be too afraid. Soldering a wire back on to a coil is something that is done quite frequently by pinball owners.

Bullhockey. No guarantee it won't slide off once the wire is broken. Heat shrink tubing stays put.

If you want the tube, clear rubber tube. Hardware store, hobby shop, aquarium store, etc. etc.

LTG : )

It's meant to insulate the lug from accidentally shorting out on anything else. You should be fine.