14cm fan, is the height matters ? or standard height ?

Fan speed alone is not the only factor that a fan can bring how many airflow.
Just go any lowyat shop and ask for options. I won't be too concerned on speed though, anything with 1200rpm should be ok from safe side.

Also check whether your old fan is with 2 cables or 3 cables. 2 cables is either unregulated or voltage regulated, which 3 cables is PWM regulated. Better if you buy the matching variant (2/3 cables) since PWM fan doesn't work well under low voltage.

You can just connect the wires and wrap them around with wiring tape if you don't want to solder. It reduces the risk of getting electrocuted during soldering. Wear a rubber slipper when you work on it.

At this point of time, I don't even know the PSU even bother to read the fan RPM or not. However, indeed the third cable usually is for speed reading if talking about conventional PC fan.

then you should be able to direct swap them
i wouldn't worry about the bearing type
new fan bearing vs much older psu
high chance the bearing will outlast the psu

ic, but i think psu temp has no effect since the fan will be blowing towards psu and out
unlike the old design or sff psu where the fan sucks air from psu

You have few options actually
First option just install the pwm fan or 3 pin fan to your motherboard header or fan hub and the wire actually long enough exit from the original bulky wire hole then control the speed via motherboard bios or vise versa

The other options straight forward install a 2 pin fan cut off the header reused the old fan headers if needed

The other options 3 pin fan you need to loop the yellow and red together or the spinning speed just too slow, nowadays you hard to find color labeled 3 pin wire fan mostly black so you will need multi meter to test which is 5v and loop The other 2 together

2 pin red and black usually noise level quite loud

If you don't want so much hassle just do this example below

sorry but why introduce more hot air into the casing instead of sucking the existing air out?
i still think blowing air towards components need to be cooled down works better than sucking the heat from those components
the only drawback would be dust accumulating on the components