10baseT can run 150-200 meters on CAT5 cable--some vendors even document this as being acceptable.
I've never seen anything documented to the effect that 100baseTX can run more than 100 meters on any type of cable, nonetheless if you're desperate it's certainly worth a try. You get away with what you get away with.
As for media converters vs fiber ports in the switch, if you're using a separate media converter, that introduces several additional points of failure (additional cable, connectors, power supply, etc) so in theory, with all else being equal, it should be less reliable. In practice I don't think it makes a whole lot of difference.
If the switch is well built, then I quess that switch witrh fiber ports would be more reliable. With a media converters you have more boxes in the system (media converter and switch) plus more wiring needed (cable from media ocnverter to switch, power feed for each ot them). Usually more ports means worse reliabity (more things can go wrong).
Usually with good quality copper (CAT5e CAT6) you can run Ethernet (10Base-T, 100Base-TX) beyond the magic 100 meter limit. Usually it works very well. I have read about installations where the Ethernet signal is run for up to 150 meters without problems..
Using longer than 100 meters wirings is not accordign the oficial standards. Running more than this can risk you to system become moren sentive to external noise problems causing transmission errors (those showing as packet loss typically). The rule of thumb is that if the wiring you use is of better quality than for what the Ethernet system you use is designed for (CAT3 for 10Base-T, CAT5 for 100Base-TX), then your cable has better shielding properties and lower signal loss, so you can run lengths that are slightly longer than the original 100 meter meximum length without noticeable problems. This is not a recommended practice, but it seems to work both in theory and practice.
It might be that Cat-6 or, if it is now available, Cat-7 could get you a little farther, but probably not much. If you can meet the spec., end to end, with a good cable tester it should be fine. Use a switched net so you don't have to worry about timing limits. I would look first at the high frequency attenuation, 100MHz or so. It is said that crosstalk is also a problem, but that is mostly because the received signal gets too small.