Long sleeves for safety? I took @Jason's and @Neil's advice and practiced wearing clipless pedals in cold weather. I fell three times (nearly four) last week, all at very low speeds: twice because of ice, and twice because of having my shoes clipped to the pedals and not unclipping in time.

Falling didn't hurt: at all. At the time I was wearing a short-sleeve shirt, and a long-sleeved uninsulated Gore-Tex wind-and-rain-proof anorak (and jeans and gloves and a hard hat). I bruised my hip slightly, one of those time; never any cuts or abrasions.

The weather is getting better (it's above freezing, this week). Eventually, it will be hot. I commute and I don't know about specialist bike clothes. I expect I might wear, I don't know, summer shorts and a T-shirt in hot weather (and shoes and a hard hat and maybe gloves).

Ought I, for safety, to wear long sleeves or something, even in summer, in case I come off again? I can't see myself wearing a wind-proof anorak in sunny summer when the temperature is in the +10C to +25C kind of range.

Should I have something though, to protect my arms if I fall? Would long sleeves make a difference? Is there a fabric that's protective yet cool?

I don't see myself wearing armour just for commuting. On the other hand, my fear / worst case scenario is that I might fall, leave bits of elbow tendon behind on the pavement, and then people say "what a fool: didn't he know he's supposed to wear XXX for, you know, basic safety when cycling?"
Unless you're out derbying or have a complicating balance issue I wouldn't worry about it.  Falling off a bicycle when commuting or recreational riding should become an extremely rare experience with practice.  Tank tops and shorts are perfectly acceptable bike wear in the summer.

Ice is of course a different beast; the best advice I have for ice is that you should be prepared to fall and do not have anything in your pockets that you don't want embedded in your leg.