You wouldn't use screws with linoleum, you'd use a glue. Linoleum is a great product as it is made from linseed oil and it actually 'heals' itself when cut. That is assuming you are really using linoleum and not vinyl. When we started exploring replacing a floor we asked about linoleum because it doesn't offgas, and is an eco friendly alternative. We found that flooring companies often called vinyl (which does offgas) linoleum, in fact they would call it linoleum vinyl! Of course this whole paragraph is about flooring not in a cage
I don't know how well this would work in the bottom of a cage. I too have considered it in the past and a possible idea for the bottom of a C&C cage and I'm just am not comfortable with the idea yet.
So what type of wood cage are you planning? Wood frame? With what type of sides, top and bottom? If done right a wood cage could be really nice. Yes urine getting soaked into the wood is a concern, but if addressed and correctly done it could be a non-issue. Standing Bear at the Flash and Thelma has some really nice cages in the hedge house made of wood (
http://hedgieflash.org/rosie.html scroll down the page). I've also wondered if the PVC lumber they sell for decking and fencing would work for a cage. I haven't explored the idea much though.