Buying a car will hurt your chances. Talk to a loan officer or mortgage broker, who can tell you whether the impact would be minor or significant. A lot will deal with your debt/income ratio. If you're earning $1,500 a month, it'll hurt a lot; if you're earning $7,000 a month the impact will be far less severe.
Also, talk with an accountant about whether putting 30%-40% down on the rental is the best move. It'll certainly reduce your payments. But will you have enough liquid assets after the purchase. Also, if you're expecting the property to appreciate in value (and you should; you shouldn't be buying a property you expect to lose value), you may want to increase your leverage a bit by putting less down.
Hope that helps.