Exam

You should be between 3k to 5k to get admission in IHM-Mumbai in 1st round. And for 2nd round, you must be between 5k to 7k. I suggest you to not join IHM-Mumbai or any other hotel management course, better do simple B.Sc.
I am writing this because, I got IHM Mumbai in 2017 in 1st round of counsellin

Hi anushka some colleges have entrance some not it depends on their choice

What is your exact question? What kind of guidance do you need? Please be clear.

No. CAT is one way to get in MBA colleges. You can give MAT XAT GMAT ATMA CMAT IIFT NMAT etc. Many options are there. Don't limit your options only to CAT.

Go for online class options. You can see whenever you want and can study from wherever you want. With an ease. And can clear doubt from n no of videos and you can do it for free or paid also. See as much as YouTube video of different professors for your CAT preparations. All the luck to you.

If internship is of not more than 2 months, surely go for it, it will help you gain good practical knowledge as well as something good to write in your resume. For GATE, I think, if you study well in the rest of the time you have, you will surely crack GATE with good marks.