Just out of curiosity, is it possible to equip a tune o matic bridge, and in place of the stopbar, have a floyd?

The Jazzmaster is similar but it's tremolo is different.
I don't even shred
I was looking through this book and found an old squier bullet, it had a standard tremolo and a TOM Esque bridge right in front of the tremolo.

But thanks anyway.

And a second question, would locking tuners/locking nut on a jazzmaster keep it in tune?
I don't even shred
I'd like to see that picture of the Squier.

There are other options you should try before buying something expensive.

There's several factors, when you're trying to keep a guitar in tune:
- Stringing technique
- proper setup
- As little friction as possible.

So let's attack these possible problems.
Friction - Your guitar has what are called string trees:

The circled metal thing keeps the string from popping out of the string nut. Useful, but they also create friction that prevents the string from returning to its proper pitch after a bend or something. So...apply some sort of lubrication to to the underside of the string tree, as pointed to by the arrow. Use something like Big Bends Nut Sauce, or Guitar Grease. Even pencil lead will work. Simply grind the pencil lead into dust on a paper, and use some of this dust on the point of friction.
The same problem occurs at the string nut. Each groove has a bit of friction. So you can apply the same lubricant there, underneath the string, and that will help. You can also buy the appropriate replacement nut, but the nut should be made of graphite. There's less friction from that material.

Moving on to setup. Jazzmasters are kind of awkward guitars, and need to be setup properly. You can do it yourself, or bring it into a shop. I don't have any experience with working on a Jazzmaster, but I can point you towards some helpful links:

Lastly, stringing technique. Yes, it's possible to string a guitar improperly. I actually mess with my Strats trem a lot, and it stays in tune really nicely. Of course, I have it lubricated at the proper points of friction and have a graphite nut, but I don't use locking tuners or anything like that. The secret? Stringing your guitar the right way! Here's a link that will show you how I do it: