hiii ,

yes , if you take mechanical engineering you can apply for all the automobile companies in india and abroad and you can also go for masters in automobiles for further studies almost all the thing in automobile is also taught in mechanical engineering .

for colleges you can get through jee mains and for branch you get in each college use the free jee college predictor - https://engineering.careers360.com/jee-main-college-predictor?ici=cp-listing_engineering&icn=JEE-Main-College-Predictor