If the slave cylinder bore and piston look otherwise okay, I think I would be tempted to go for the seal kits too. The reason we don't bother so much over here resealing and reconditioning parts is that seal kits aren't much cheaper than complete aftermarket replacement parts.
Personally, if the engine oil hasn't been in too long, I would flush all the fluid out NOW, and replace or reseal the slave cylinder, see how you go. Most modern rubbers are pretty resilient to oils, short term. If the engine oil has been in there a while, you may well have problems though, as you say.
The clutch damper is only there to soften the pedal feel and sometimes to prevent clutch chatter, particularly on cars with this type of external slave cylinder. Normally, you can do without them quite happily, and some remove them intentionally to take some of the mushy feel out of the clutch pedal. I haven't had this Jazz long, and to be honest, I didn't realise the Jazz had one.
There are various types of damper, but the Jazz one looks like a sprung plunger type, so there will be piston seals or a rubber diaphragm in there. They are simple devices though, you should be able to dismantle it, clean it and check the condition of the rubbers. If all else fails, try removing it altogether, or just remove the innards.
