Can botox injections help get my drooping eyelid corrected permanently? If so what will be the probable cost?

Probably not. Botox will temporarily relax a muscle. If the eyelid droops, you might need a ptosis repair. Please seek a consultation from an opthalmologist specializing in ptosis repair or a similarly qualified plastic surgeon.
No. Botox can't correct a droopy eyelid but it could make it worse. Drooping lids that are not related to skin excess usually require repair of the levator aponeurosis which has torn or loosened with age. See a plastic or opthalmologic plastic surgeon.
Not likely. Botox/dysport can selectively weaken the brow depressors. If you eyelid is closed from blepharospasm, Botox can temporarily correct this. Repeat treatments are needed to maintain the results. If you have significant ptosis of the eyelid, Botox is not likely to help. This usually requires surgical intervention for correction. Seek an eyelid specialist in your area.
Drooping eyelid. Botox wears off after a few months. Surgery is a more definitive option. Have your eyes evaluated by a board certified surgeon.
Sadly NO. Because the length of time botox/dysport type product effects usually don't extend beyond 3-4 months, there is sadly no permanent fixes with these chemical "active muscle movement inhibitors" , and they really aren't fda approved "cosmetically" for persons under age of consent either (warning, if you bring your parent to the Botox consultation- they may wind up taking your medicine!).