Hose cleaning is fine for routine maintenance of the filter because most material will be simply stuck on the media surface. When you are doing a plaster startup, there is a ton of microscopic fine calcium carbonate particles that are generated. These particles can get embedded in the filter medium and basically act like a plug. I have seen some pool builds where the builder is courteous enough to use a pair of old cartridges for the startup process and then, after the 30 day period, install the owner's original cartridges so as to not foul them up...it's rare, but I have seen that before.
To chemically clean cartridges is not an easy task but it can be done by the pool owner. It's a 2-step process where you first soak the cartridges in a cleaning solution designed to remove any organics (oils, bather waste, hair, skin particles, etc, etc). Then, after you rinse that, you soak the cartridge in a diluted acid solution to dissolve any embedded calcium. The organic clean can be done with either a TSP (trisodium phosphate) solution OR with some automatic dishwasher detergent. I prefer TSP as it is simple to do and I can find it easily in the paint section of my hardware store (just be careful because there are many "green alternatives" to TSP that don't contain phosphates and yet are labelled as "TSP"...weird...). The acid soak is done with a fairly dilute acid solution, typically 9-to-1 water-to-acid ratio. You can soak all four cartridges at once by using a large plastic outdoor garbage can and putting a thick, outdoor trash bag in it. Juts add the cartridges and the cleaning solution and then tie up the bag so that everything is submerged and let it sit for a few hours. Rinse and repeat with the next solution (use a new trash bag tho...).
And please, DO NOT do this anywhere near your nice new patio....TSP will corrode the heck out of anything aluminum and muriatic acid will etch stone and concrete. Simply go to a part of your yard where it doesn't matter if you release the cleaning solution, preferably over a dirt pile. You can always hose it down to dilute everything but, honestly speaking, the chemicals will have little effect on anything given the dilutions used....
OR
You can just drop ~ $400 on new cartridges.....
As for your heater, I doubt it's damaged at all. You would have to have run it while the plaster was curing. As I said, I don't think a 50% increase in pressure is normal for a heater but I have no in-depth knowledge of them.