MCU card design considerations
Is it enough to place only one bulk 10uF/6.3V capacitor (tantalum), instead of 4 of them? These little buggers are quite expensive


Depends on the current needs of the chip, but you don't have to use  tantalum... You can use 10µF ceramic, or aluminium electrolytic, thru-hole or SMD, although they will be taller than tantalums, so that may be an issue.


  Is it OK to place traces under a crystal? 


It's... meeehhhh....

If you are short on space, why not use a SMD crystal instead?


  The dsPIC has power supply pins on all 4 sides. Is it ok to connect these pins under the microcontroller?


GND goes to your ground plane.

You can use traces for VCC, it is wise to put a decoupling cap on each power pin. They can be under the board.


  I plan to use a combination of 100nF/1uF ceramic capacitors between Vdd and Vss, placed on a bottom layer. Is it enough to use only one pai


Is this a personal project? If so, you aren't trying to save 10 cents. Put 1µF MLCC on each power pin and forget about the 100nF. ESL of 0805 caps varies very little with capacitance, so 100nF would only be interesting if it allowed a much smaller package, which doesn't matter anyway since the ESL of your chip's pins is high. So, just stick a 1µF cap on each pin...

Check the stability of your LDO with the chosen caps.

And... Add a few GND pins on each connector. Like 4 per side at least. This should keep your module from becoming an antenna if you mount it mezzanine on another board.