Yes, a rubber washer is included
I believe you can from the manufacturer or the distributor. You need to ensure that the fit to the bottle is correct. I came across two different openings. The depth of the pump stem can be shortened as needed.
Standard size drilled for a 3 hole sink faucet. I had 3 holes from an old faucet. I used the middle one for a single hole faucet and the two outer holes for soap dispensers, one for dish soap and one for hand soap.
Please contact Design House Customer Service at 800-558-8700 for best assistance with your replacement part inquiry.