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Handle well is subjective unless you will be racing. The 50 suggestions of FWD McPherson strut cars should probably be crossed off if you truly care about handling.
So is "good mpg"
E36 isn't a bad suggestion. You can get into some E46s for that coin (higher mileage).
You could probably also find a bugeye WRX around 6k if you look hard enough.
Best fit from what I can tell: Integra GSR.
Depending on the exact model you choose, an SN95 will probably have a cheaper purchase price than an LS1 F-body, but will be more expensive to modify to the same level of performance, IMO.
As for weight: when I began looking for an LS1 car, I weighed a stock, 6-speed, T-top Z28 and it was 3333 with the T-tops removed. (Anyone else bring wheel scales when you go car shopping?) I don't remember how much gas was in it. My almost fully prepped ESP Z28 is a '99, hardtop 6-speed and weighs 3365 with 1/8 tank, but that's with 17x11s, 315s, SFCs, big sway bars, and other things that bump up the weight a bit. I can probably shave 40-50 pounds off the car with a smaller battery and lighter clutch/flywheel setup.
Mileage is in the high 20s. In 6th gear, the car only turns 1500 rpm at 70 mph.
How about instead of a 5.0 + Volvo, a 2.3T + Volvo... very useful, and very fun:
