JEE

I can tell you the fee structure of Kota's coaching.Fee structure for JEE is around 1 lack 10 thousands per annum and for every coaching centre and and around 90k per annum for NEET.

The college takes admission through 3 processes:
1. JEE counseling
2. MHCET counseling
3. Institution seats
You require capitation fees in order to take admission with 3rd option.

Your JEE marks are good. I think you should apply for re-evaluation of your 12th paper since a student scoring 133 marks in mains generally have a percentage between 75-100.

You can try for low rank NITs on the basis of JEE Mains paper 2. Try out for NIT Hamirpur, NIT Raipur, NIT Patna. You might get a seat in one of these colleges.
Other than these, there is Chandigarh college of Architecture, Nirma University and similar government institutes that you can go for.
I hope

It depends on your category i.e. general, SC, ST or OBC. If you belong to general category than it's quite impossible to get into Mumbai but if you belong to any other category, than be sure to get admission.

Yes, you can. But it is better to drop than to study in college for JEE because you will not be able to study for JEE whole heartedly.

Your first preference should be to concentrate on 11th and 12th examinations (without which all your efforts might become futile).
Parallel to that you can start preparing for the joint entrance examination (JEE), by following meticulously the following steps:
* Have at least four to five years' quest

As you live in Gujarat, you need to get a score of 200+ to get admission in a good reputed college in Pune like DY Patil College or College of Engineering.