HPI Wheely King IMO

1/12th scale, but in actuality its about the same size as a 1/10th scale Traxxas stampede.
Pretty much the only way to break the sucker on stock power is drive it off a sheer cliff as its plastic is VERY flexible, and it has giant ass tires to absorb most impacts.

I will only say you will not get big jumps as it only goes around 15mph, and you really can't upgrade its motor to brushless(what you need for big jumps) as its transmission can't handle it(well without a pile of modifications)
But in that price range you pretty much won't get anything that can do everything you want.

Breakdown
dirt roads 2wd, or 4wd will be fine, and any scale Tires more make the difference here

grass IMO 4wd, but 2wd can do it if its very well maintained, and the grass is short. Vehicle type depends on scale. 1/12, and 1/10th MT's work best, 1/8th truggies, buggies, and MT's have no real issue due to their weight(1/8th is way out of your budget)

jumps basically any, but it has to be fast

Woods Trail truck, scaler, crawler, or a slower solid axle MT will deal best.