1- to drop jQuery UI optimization, just enable 'jQuery' but not 'jQuery + jQuery UI' (first parameters of the plugin). As simple as that...

2- maybe the payment plugin requires specific jQuery v1.9 functionnality. Then select version 1.9 (and add a sub-version number 1) in jQuery Easy instead of 1.8.

If nothing works, leave out everything I said. There is no point on trying a remedy if it is worse than the illness...
(French proverb).

Well it is a hit or miss sometimes. There is no way I can control the way your site is going to behave. I told you what I would try and what I could see could be improved.

I am still convinced one version of jQuery can be eliminated but you would have to make the changes in the plugin and let me see why some extensions stopped functionning (I need to be able to see the errors).

There is really not much chaos on your site actually, you should have seen some sites...

One thing you should always try first before using jQuery Easy is to disable jQuery from the extensions' parameters (look if they give you such an option).

Please try and use the 'strip remaining script' option again and let me know when you have done it. I can look at the code then and see what does not work and why.

Well, one option is to make your site under maintenance for a short while. You definitely cannot do this kind of tweaking on a live site.