I was approved by Westlake Financial. how do I know which cars can be purchased using Westlake?

Ask Westlake.

I was preapproaved , there is a 2015 mini I would like to buy that is listed @ $15790, but westlake says I have to contact the dealership, that don't have any idea about Westlake, can I make an offer and how do I know what I have I need down and the monthly?

I suspect these "preapprovals" are little more than an effort to get you to use Westlake. When you contact the dealer they probably will then apply (on your behalf) to Westlake or whoever they use for financing. Negotiate the terms of the loan with the lender or at the dealer.