These various boilings serve different purposes.
The jam mixture is boiled to reduce the liquid and bring the pectin to its gelling temperature.
The bottles and lids are boiled to sterilize them as you said.
The final water bath kills any bugs that were introduced while filling the jars. Additionally this boiling ensures a good seal on the bottles. In the time it takes to fill the bottles the jam will have cooled somewhat and you might not get a good seal.
Killing bugs with boiling is not an instantaneous process. It takes time at those high temperatures to kill a high percentage. If the bottles aren't pre-sterilized then you are putting yourself at a disadvantage as those bugs would be distributed through the hot, but no longer boiling, jam.
Of course, all of these boilings will do some sterilization and are somewhat redundant. It's all about reducing risk. You can bottle with clean but unsterilized bottles and if you work quickly you can skip the hot water bath. Doing so will greatly increase the chances that one or two bottles might not seal or that some bug might survive and spoil your stuff. Reliable sources indicate that if bottles are not presterilized that the bottles should be processed longer in the water bath.
Anecdotally speaking- I have in the past skipped the water bath when my jam was very hot and acidic. I would usually have one or two bottles out of 20 that wouldn't seal. I no longer skip the water bath and I almost never have any bottles that fail to seal.
I have never had jams go bad but this is more luck than proof.
Pressure cookers and pressure canners are the same thing; with the canners being larger, and often having a pressure dial. Both can reach the same pressures and therefore temperatures if designed and manufactured correctly
The pressure canners dial gauge is more accurate for adjusting for food types and altitude, as you can get exact numbers not just 10 psi and 15 psi as with typical weight regulated pressure cookers. But for low acid foods you need 15 psi anyway
Pressure preserving is done because of the higher temperatures reached. This saves time and energy, and some food react better to a short 250°F (120°C) cook that an long 210°F (100°C) cook
Pressure preserving high acid food is worth doing as long as the food does not deteriorate quicker due to the high temperature (some soft fruits will)
For preserving low acid foods you need a known good pressure cooker that can reach 15 psi (103 Pa) and therefore 250°F (120°C)
With a commercially published recipe that has been acid level tested, you will have a published time that will be safe. For a home-made recipe you have to take the worst case scenario for the acid level of the main content, and cook for that. Your local government health department will most likely publish tables for this that you should be using. Remember to use altitude adjustments for timings too
Just relying on adding a teaspoon of vinegar or lemon juice per jar is not a safe and scientific method for raising the acid level, though many people do it, and have not come to any trouble with it?
You pressure cooker should come with a manual that confirms what it can or cannot do. Download a new copy if this has been lost
Also see What kinds of pressure cookers are there, and what are they good for?
Best Answer
Yes, basically anything cooked will last at least a few days in the fridge.
If you want it to safely keep it longer than that, you can freeze it. It's probably best not to try to reprocess the jars though, since you don't know exactly how much water got in (and how much acid and sugar got out), so if you're unlucky it could upset the recipe enough to make them unsafe.