Can you put another airline's FF number when booking flights (ie.

Can you put another airline's FF number when booking flights (ie. VA FF number when booking through DL?)

Depends on the airlines and their alliances. E.g. you can book DL and put VA. 

If in doubt, just do a dummy booking and navigate through to the passenger details form. 

I booked a Cathay Flight on the Cathay website a few weeks ago, put in my Qantas FF number.  Paid the flight, logged off, and logged into my profile in the QFF website, and my Cathay itinerary was listed under "My itineraries".  So i would say Yes.  Probably as tronixstuff said, it has got to do with alliances, and whether the other airline has any code share agreements in place.

Yes it is possible assuming it is your own account and not someone else's.

In an institutionalised alliance such as OneWorld, that is a given. Booked on CX, QF & BA using AA frequently flyer number. In all instances not only is the number recognised but the status is also recognised, facilitating seat selection and additional baggage allowance.

However in a non-institutionalised alliance like VA and DL, while the ability to record the VA number on a DL booking should not be a problem, I am uncertain if the applicable additional benefits are also recognised. Always bring your frequent flyer card and present it to the check in agent during check-in (even if you have checked-in online and only want to dump your bags). Also keep your boarding pass until you see the points/miles accrued to your account. It may facilitate the escalation of a problem if the points/miles are not being accrued upon follow-up.