Drilling into wall with damp proof membrane

Yes you can, but you need to use the correct fixings, cavity wall fixings with seals, to stop moisture and salts migrating through, try platinum or newton suppliers Good luck Joseph

Yes this won't be a problem. Use a masonry drill bit

Yes, wall fixings can often be installed, but the method depends on the membrane system fitted. Drilling through it creates a puncture that must be sealed correctly to prevent moisture reaching the new plaster. For brick or block walls, use a masonry drill bit sized to suit the specified wall plug. Start gently with hammer action switched off through the plaster. Heavier items need fixings anchored securely into the masonry or suitable structural supports. Before drilling, check with the original installer for the manufacturer’s approved fixing and sealing method, and whether this affects your guarantee. Ordinary wall plugs and household silicone should not be assumed suitable. Also check for concealed pipes and cables.