It is not recommended to attempt modifications to your instrument without knowledge, practice, and the correct tools.
To answer your questions:
You can drill safely through the plastic used in guitar bridge construction. Carbon fiber and composite graphite can also be drilled, but more care would have to be taken to avoid splintering or chipping.
Standard multi-purpose or wood bits will work.
to get the correct angle, extra long drill bits, often used with an extender, may be used. The top would need to be protected from the bit and extender when drilling, as you need to drill across level to the face.
It is good practice to put in a stop plate or sacrificial piece of wood on the other side of the holes to stop tear out and slipping through to damage the bridge.