As a general rule, male bettas are a bit more brightly coloured than females, although females are very pretty too. Another method of sexing is to look at the finnage: in males, the fins are usually longer. I say 'usually' because in bettas with plakat finnage (also called short-finned), males have short fins also. Males make bubblenests and flare too, although some females also have this behaviour. Not being very useful, am I? lol
In answer to your second question, after the male and female have spawned yes, you should remove the FEMALE ONLY, for her own safety and for that of the eggs. You should leave the father in, he is crucial to the spawn because he maintains the bubblenest and cares for the fry. Here's a link to the spawning info on Bettatalk.com (great site ):
The most reliable way to sex bettas is to look on their belly right behind their ventral fins. If there's a white spot, that's the egg tube of the female.
Example:
Males don't have this. Also, if the fish has long flowing fins, it's almost definitely a male. However, this doesn't mean that all bettas with short fins are females, because as SPLiSH mentioned, plakat males have short fins.
When spawning, the female is removed as soon as they're done laying their eggs. Generally the male is removed as soon as the fry become free-swimming, but some breeders choose to leave him in longer.
Betta splendens pairs cannot be housed together permanently and are only combined during breeding, so how often they reproduce is up to you. Generally it's a good idea to let a pair rest for 6 weeks between spawning... and since you usually get 100+ fry per spawn, it's probably a good idea to let you rest too!