My tire's max PSI is 125 ... I have it filled up to 125 to prevent any more pinch flats (had two in a week!). Now when I ride my bike I can literally feel every bump in the road because the tires are so firm. Is there anything I should watch out for now? Or am I basically pinch-flat proof now? Should I be worried about the high pressure at all? I don't want to end up like this guy ...

https://www.youtube.com/watch?v=enRKvjKr7l0

Sounds like you really need larger tires. My undercarriage would not be happy with 125psi tires for long.

Your bike's undercarriage or your biological undercarriage?

Biological.

The most pressure I run in any of my bike tires is 80 psi, and judging by the lack of pinch flats since 2009 or so, that's enough.

I run my road bikes tires at 120, and they're noticeably harder than what little guys run at 80-90. That being said, I dont have pInch flats.

When the road gets wet, though, the risk of sliding or skidding seems greater.

Fatter tires.

Would you recommend putting 700c/25's on the rims (which are designed for 23s) instead?

+1 fatter tires; simply better in every way. Or I can refer you to a good dentist. 125psi does increase risks of rim sidewall failure a little, but with regular inspection is not overly concerning. The failures I have seen were on over machined sidewalls on very lightweight rims like 90s era Mavic open-pro. Haven't seen one fail since the 90s either.

25's aren't really noticeably better. I was 175lbs when I finally wisened up to try out 28's. That's my ideal road tire size for healthy me. Now that I'll likely be up around 240 or more this spring, I don't think that I'll he on tires smaller than 32mm. I've used 35's on velocity deep-v model rims with zero problem.

Measure the inside of your rim.

Scroll towards the bottom, there's a section on tire width.
Tire Sizing Systems

I have a bike that can only fit 25's. It's not worth what I paid for it so I'll be putting 650b rims on so as to be able to fit a more sensible sized tire.

23's are sooooo passe.

I'd try a tubeless setup with 25's.

Riding style has a lot to do with pinch flats as well. Unweight the bike or even give it a little hop over sharp edges that may cause pinch flats.

FWIW, I'm 165 and don't get flats on 23mm tires at 100psi. Haven't tried any lower pressure, comfort is good using supple tires.

+1
Supple Tires, Supple Rider.

My weight tends to fluctuate between 215 and 230 lbs.
I keep my 25mm tires @ 120 psi rear and 95 psi front on the Sporty Quick bike, as I often carry 10 to 15 lbs. of 'gear' in the Topeak rack trunk w/ the fold out panniers on my 30+ mi. r/t commute, often stopping for groceries on the way home.
My experience has been that the softer front tire improves comfort (mutes 'road buzz') and handling (improves traction), while the higher pressure of the rear tire eliminates pinch flats (tough to 'hop' rear tire when loaded w/ 20-30 lbs. of vittles in the panniers).
I learned long ago to 'Ride Light' (unweighting saddle, using legs as active suspension) even when I was much heavier than I am now.

Sure. I'd go up to a 28mm tire if your frame/fork has enough clearance.

HvPnyrs, Could you explain what you mean by using the legs for active suspension?

If HvPnyrs doesn't reply, I presume it means standing on the pedals as you go over a bump to shift more weight onto the pedal axle and off the rear axle and tires, lessening the force on the tires that might produce a pinch flat, as I understand it. I do it myself after seeing expert riders do it early in my riding experience.