The shield should be connected only to one end to prevent ground loops
Yes, let's go to the bother of buying shielded cable, and connecting up one end, then injecting exactly 100% of the noise we hoped to block, at the conveniently-opened end!
Yeah no,
always ground both ends. A galvanic ground isn't required, but an RF ground is. If you have ground loop problems, use capacitors. Multiple capacitors, stitching around the shield, to maintain RF shield integrity.
The problem with shielded wires is the capacitance from the signal to ground is higher, so longer runs bring new problems. Ethernet is ok for pretty long runs but likely not so ok for such long runs with shielded wires.
Uh?
More capacitance would imply a lower characteristic impedance. But the manufacturers know this, so they increase the insulation diameter to compensate.
At best, velocity would be lower, due to the higher fraction of dielectric (less air space). Loss might be higher by the same amount, or it can simply be made with better (bigger?) wire.
When you buy CAT6 or whatever cable, shielded or not, and its datasheet says so-and-so impedance, velocity factor and loss... you're getting what it says. That's simply it.
I never have an issue with ethernet wiring so far, I found myself in the first one not so long ago, I asked around and it was an old wire that was placed in a different place, I made replace the run and all good. Twisted pairs are pretty good rejecting noise, even better with the fully differential and balanced scheme used in ethernet, so the shielding is not as critical as it is in other signals like analog audio where balanced is not so balanced, differential is not so differential and environmental noise inside the band is much higher, like power lines running inside the band of interest.
Ethernet itself is extremely robust. OP appears to be more concerned about Ethernet radiating noise in a very sensitive environment. Note that Ethernet transformers are only rated to 30 or 40dB worth of CMRR, which means, out of a ~1V signal level, a good ~10mV of noise might be expected, at unlucky frequencies (usually switching edges). In practice, it's better than this -- as it must, because most commercial EMI/RFI regulations are in the single or sub-mV range.
Note that shielded Ethernet isn't particularly useful unless your entire lab space is shielded. Ambient sources of noise: computers, power supplies, phones, commercial radio, etc., are all in the same ~mV range, and aren't coming in on wires. There's only one way to block those.
Tim