You're not really going to find any significant differences between .22 LR pistols. I do believe the Thompson Contender can be chambered in .22-250, and it's a very simple single shot mechanism... however, probably not really what you had in mind.
What kind of loads are you going to be putting through it? You really can't go wrong with a Ruger Mk. I/II/III, except that they can be a bit difficult taking apart and putting back together for cleaning. The Sig Mosquito is an excellent pistol, but very fickle about which ammo it likes. Of course, a good revolver is always an excellent choice for barrel length and reliability - but only from reputable manufacturers, such as Ruger, Smith & Wesson, etc.
However, I can't really recommend a .22 for self defence. Granted, shot placement makes all the difference, but still...
it is a little less powerful but not by much
Depends on the application. If you mean power, the higher the caliber, the more powerful it is. for example, a .22 caliber rifle is much less powerful than a .50 caliber rifle.
one the size of my D.i.c.k
Depends on which 45 and which 44 chambering you are talking about the load
It depends on the load
yes
Really depends on the load, but the .270 is typically the more powerful of the two.
.950 J D Jones
trade it in for a 45 caliber
No. There are many low-quality pistols in .380, but this is not related to the caliber. If it is a reliable model, it won't jam more than any reliable larger caliber. Badly designed and sloppily made pistols will jam, quality pistols will work perfectly. For instance, SIG Sauer P230 is chambered in .380 and is one of the most reliable pistols out there.
Reliabilty is a function of the mechanical aspects of the weapon, not the cartridge.
No, not by a longshot (no pun intended). Almost all "major" caliber handguns are more powerful than the 9mm.