They hate it because they make money and are incentivized to find financing for you and will make money on the trade.
yep. they get a kickback from the financing company, so you have the best chance of getting a good price if you take their financing.
personally, i've found that manufacturers' financing rates are generally better than banks, but even if you think you're going to get a better rate from a bank, you maximize your negotiating leverage if you take the dealership financing up front and then later pay it off with a loan from a bank.
if you haven't bought a car from a dealership before, the end of month effect is real. know what you want to pay and go in on the next to last or last day of the month and tell them your price. the first dealership might not take it, but someone will. can also do it all by email.