Your Dutch licence is valid to be used in Spain until it expires, or if it has no expiry date or is valid for more than 15 years, for two years after establishing residence in Spain. If it's a standard EU photo type licence valid for 10 years then there's no need to change it until it expires. Exchanging the licence costs around €28.
Regarding the second question, it's not quite as you describe - there is no A1 licence granted, however after three (not two) years - see the PDF below - of holding a B licence one can drive motorbikes up to 125cc within Spain (note that you can't ride a 125cc bike outside Spain unless you take and pass an A1 test). My understanding is that any EU B licence would give you the same rights - an ex-work colleague used to ride a 125cc scooter on a UK issued B licence - but I've not been able to find a government source for this so far. Given EU law and rights etc, and given that if you were to swap your Dutch licence for a Spanish one (and held B for at least 3 years) you'd be able to ride a 125cc bike, I don't believe there are any issues, but I'll continue to look for a source. To be 100% on the safe side, it's worth swapping the licence, but in general as long as you can get insurance you should be OK.
At the bottom of page 8 of this pdf, it doesn't mention that the licence must be a Spanish one in order to drive a motorbike on a B licence (the document is aimed at Spanish learner drivers, but is the only official source I've found so far that says that it's valid).
(As an aside, the law says that the bike must be below 125cc, and with a maximum power of 11cv and a power/weight ratio of 0.1 cv/kg (just in case your bike doesn't meet all three of these requirements)).