A friend of mine bought a '92 FE350 new and still has it, although it hasn't seen much use for the past 10 years or so. The only work on the motor he had to do was to replace the valves (he didn't realize they needed periodic adjustment - they got checked for the first time after I bought my new '96 FE501 and asked him when the last time he checked the valves on his). The main thing to remember with these bikes is to change the oil very frequently as the oil capacity is less than a quart and there is no oil pump or filter - just a screen you should clean each time. My friend Jeff and I changed the oil on our Husabergs after every ride and never had any reliabilty problems. I owned my '96 FE501 for 8 years and did nothing to it other than routine maintenance. I did a lot of trail riding, some desert riding, and an occasional cross-country race.
As far as the bikes being suitable for use as a dual sport, what you need to remember is that these bike were developed for competition and are very uncomfortable to ride very far on the road. I originally had the idea of using my '96 FE501 as a dualsport but changed my mind very quickly after riding it on the road a bit. Changing the oil every 100 miles would get a bit old, not to mention putting up with the vibration. My Suzuki DR-Z400 makes a far nicer bike to ride on the road, but off-road the Husaberg beats it hands down.
I'm not too sure about parts availability for the FE350, I do know that '96 was the last year they made a 350. Most of the motor parts should interchange with the 400cc motor but the piston is smaller (86mm vrs. 92mm) but the stroke is the same. All the other parts should be the same as the 501cc motors. The advantage of the 350cc motor is that it puts out quite a bit less power than the 501 motor so the parts are not as heavily stressed and less likely to fail. The SEM ignitons can be a bit unreliable and there was a problem with them failing from heat in that era but that was mostly when raced in motocross (I remember seeing pictures of do-it-yourself igniton venting in magazines). The good thing is that replacement ignitions are available from se3vgeral sources.
The main reliability issues of the early Husabergs related to the valve train and main bearings, But regular maintenance (frequent oil changes, check the valves every 10 hrs.) will keep these bikes running a long time without needing to rebuild the motors if you are not racing them.