The best way to travel south is to take an 2AC class berth on the train. It will not be any cooler down south. YOu could go up to Uttaranchal or himachal pradesh if you want to escape the heat.
Where are you staying in Mumbai? How much is it? What's it like?
I"ve not been to Goa. Kerala, most people visit kochi. There are many guesthouses. In Alleppey, you can find local homestays by going to the tourist info office in town. I would avoid the ones along the waterfront. they are probably fine but there are more authentic ones further out.
There is a good place listed in LP guidebook if you look under near/around Alleppey. I would highly recommend this place for a relaxing time. It costs 1000 per day all meals included and the food is good. Its on the backwaters and you can avoid the boring boats by doing this. YOu get there by govt ferry.