Working out for MMA is MUCH different than working out just to get a beach body. While having low bodyfat is good, if you plan to stay at your current weight you better be pure muscle.
1. When running, dont do it. If you run I suggest sprinting short distances like 20 yds or so back and forth with a medium amount of weight.
2. Depends, most would try to fit you in featherweight - lightweight.. its really what YOU want... you want to be huge and strong? Then work hard and get up to heavyweight. Its what YOU want and how YOU want to fight. If you like how you are now then I suggest staying in a lower weight class.
3. lifting weights is good, just dont get hurt. Also doing sandbag/suspension training is also a good workout to mix in.