As far as South India is considered, the no of people speaking hindi is different from one state to other. For example, one can survive with hindi in most northern parts of the two states Andhra and Karnataka. If you come down, not all people speak hindi, but if you try, you can atleast find one or two who are speaking hindi. This is true only in Andhra and Karnataka. As far as TamilNadu(TN) is concerned, If you speak hindi, they will think you are selling paani puri and Hindi is completely useless because nonone knows it. Since you are not going to TN, It is not a problem for you.
What is the mainstream foods i.e. what is eaten in Lunch and Dinner generally in Kerala, Karnataka and AP individually? Do they differ in these states? Is there any common dish in this region?
In general, all southindian lunches include thali (meals) with rice,sambar,rasam,curd and vegetable curries etc. For dinner, they usually eat Dosa, Idly, Parotta and its varieties. But there will be a difference in taste from one state to other . Also, there are dishes which are a speciality only for that particular region. South india is too diverse and there are different variations of food you can explore. But If you are particular about eating north indian food like rotis, you can find a north indian restaurant in almost all the major cities in southindia. In the towns, eventhough you cannot see a proper north indian restaurant, you can atleast get chappathis with some curry.
From my experience I try to get some specific regional food I encountered.
Andhra Pradesh - Spicy with lot of chillies in all dishes. Apart from thali, the biriyani is very famous.
Karnataka - Famous for vegeterian food and snacks. The food will be little milder in spiciness. For Authentic vegeterian food, you can go to a place Udupi near Mangalore, a place where Dosa is said to be invented. Since Historically Mangalore is a port, You can find different kinds of cuisines ranging from Vegetarian ,Goa(Konkani Food), Arab Food, Kerela Food and the local Tulu Food.
Kerela - Since Kerela is a place full of mountains and cocunuts and beaches, the food here is lot healthier with lot of cocunut in most of the dishes. You can get good Non-vegeterian food because of the christians and muslims living in Kerela. You can even get Beef and Pork. Even the water served in hotels will be having some spices added which will help in digestion.