Noo i don't think so that sweating can cause hair loss.. Even i am listening about it first time.. No doubt sweat is always salty but not such as great which effect hairs health
It totally depends on mode and goal which i wanna achieve but normally i do workout consist between 30 to 40 minutes.. I always do workout 5 days in a week
If you try cardio especially Yoga it helps to improve blood circulation in entire body.hence when blood circulation is in right amount toward your hairs than its mean its quite good for your hair's health as well as overall body health
You wanna cut fats as well as wanna gain some muscles so i suggest you to join Tacoma Boot Camp and Personal Fitness Trainer .. fif you wanna do it by your own than try some cardio exercise and weight lifting..