Possibility of approval of what? Do you mean to say if another employer want to file cap-exempt based on the past approval, then yes it is possible.
It takes 7 to 21 days to get the receipt for the petition. If you dont have the copy of your old H1B petition, you can request the employer to share the copy with you.
Yes, you will receive I-797 receipt with number once the petition is filed and you can track using that number.
Cap-exempt do not need the beneficiary to apply under lottery. Cap-subject has to apply in the fresh H1B quota. USCIS starts accepting cap-subject applications on 1st March every year and if selected ( via lottery because there are more applications that the H1B quota of 85000 visas every year) your employer can file the petition.