Regardless of any transistor testing - think about WHY the coil fried??

Yes, it was locked on by a shorted transistor more likely than not.

Replace the transistor when you fit the new coil AND make sure you have the diode mounted on the coil the correct way around or you will be replacing the transistor again.

Also make sure you have ordered a new sleeve for the coil as well. Some come with a new sleeve, some don't.

yes, cut it off

You should never replace one transistor in a string. The new one will possibly short straight away because of others in the string potentially being faulty.

It is good practice to replace ALL transistors in a DC connected string at the same time BEFORE applying power.

I wouldn't get too hung up on measuring them, meters all give different readings and transistors can often test OK but fault under load or when stressed.

All of these parts are dirt cheap - I just replace them all when in doubt. Sure, use your meter to quickly find a shorted transistor, that's easy - then replace all in that string and you will fix 90% of your issues.

The same as diodes across coils and switches, they are $0.01 each - I don't even bother getting the meter out - just replace them if in doubt.

I would certainly replace all three transistors and the diode at the same time. I wouldn't be too concerned about the resistors but, when the transistors are out, it's pretty easy to run the meter over them just to confirm they read close to the original values.

The 74XXX chip usually fares OK so I wouldn't replace it for a start. In any case, if the 74XXX chip is faulty it might cause the coil to either lock on OR not work at all BUT it is unlikely to take out the transistor string.

Volts DC - with the meter set to the 20V DC scale (if it has different scales).