Step 1 when it comes to car buying - NEVER 'fall in love' with a car (or if you do - absolutely do not let the salesman know - although they can generally sense it).
Step 2 is to ALWAYS take any potential purchase out for a thorough test drive - and make sure to get it up to speed on a freeway or interstate. When you come back, leave it running, and glance under the hood (and under the car) for any leaks, or noises - ideally, take it to a mechanic for a once over (get quotes - but it's usually around $75 - it's not a guarantee, but they can sometimes catch big $$ items).
As far as what to do now - since it was doing it when you bought it and they lied and told you it was unbalanced tires - I would take it back to the dealership and ask to speak with the owner. This will generally get you a manager who will throw the 'used - as-is' line in your face.
Search on Google to see if the state where you live has 'lemon laws' in effect that prevent a car dealership from selling a car with a known problem without disclosing it to the buyer. If your state does have these laws - respond to the 'used as-is' line by telling them that you want them to take the car back and issue a 100% refund.
(This will all be met with uproarious laughter on the part of the salespeople/managers - so if you aren't a particularly forceful person, you might want to bring along some violent friends to let them know you are serious).
Hopefully they will back down and fix the broken motor mount - or at least give you a reasonable discounted price. If, after your best efforts, they will not, then take the car to some mechanics and get quotes for fixing it (the dealership will be the most expensive option).
Then submit a copy of the bill to the dealership and demand that they reimburse you for the repair (this will be met again with laughter/derision). Continue to contact the dealership from time to time and demand payment (with interest) and in the meantime, post as many bad reviews of the dealership as possible, and deride them to anyone who will listen.
I bought a vehicle one time that seemed to be 100%, but within a few days the power steering pump started making noise. I took it back to the dealership and told them I wanted it fixed - for free, or for them to take the car back and refund my money. They did exactly what I described - the used car manager actually pretended like I wasn't there. After speaking with their new car manager, they decided it was in their best interest to at least take a look at it.
It turned out all it needed was to have the power steering line bled (air in the line was causing noise at the pump), but because their used car manager decided to be a jerk, they actually offered to replace the pump and high pressure line and all I had to pay for was the parts.
I declined (since I could do it for far less), and it never had a problem the rest of the time I had the car.