if you buy one with a mic plug then yes you can record on the fly
although newer models don't always offer this the old ones always used to (i have one that records live)
as for converting to mp3 or wav, you would need a PC and yes you can do that easily with the new Net MD's
i suggest checking out
i totally recomend MD over MP3 anyday i love mine and i've had it for a couple of years.
one AA recharable battery lasts me about 50 hours, i get 2-3 cd's on each disk at the equvalent to i think 128kbps MP3 (maybe its equiv to only 64) but anyway its good.
mine also has am/fm tuner (i'd totally recomend getting that)
and a microphone plug which i have used heaps.
so good luck