UPCAT

You can get government colleges like:
IET
CSJMU
Bundelkhand University
MJPRU
GEC Kannauj
Galgotias

Yes, you can get the branch of your choice and also, you can get into HBTI or IET.
HBTI is the number 1 UPTU college and IET is the 2nd under AKTU.

I am listing some top colleges under UPSEE:
Amity University
KIET Group of Institutions - [KIET]
Babu Banarasi Das University - [BBDU]
Sanskar Educational Group - [SEG]
Galgotias University - [GU]
Invertis University
Prasad Institute of Technology - [PIT]
SR College of Pharmacy

Your rank is not good to get into IET, Lucknow but you can try for KNIT, Sultanpur.

UPTU stands for Uttar Pradesh Technical University and it's conducted by UP Government. The State Entrance Examination - Uttar Pradesh Technical University (SEE-UPTU) (formerly known as UPSEAT) is an annual college entrance examination in Uttar Pradesh for engineering, architecture, pharmacy and management courses. It is one of the highly reputed exams of India top colleges under UPTU are:
HBTI
IET
KNIT
KIET
CSJMU
GEC
Bundelkhand University
Galgotias, etc.

Seats are dependent upon the college. Every branch of engineering has 60 seats. You can calculate by yourself.

No, it is not included in BCGL.

No, a rural certificate is not necessary.

No, the enrollment slip is not valid but I think you should once ask the AKTU authorities.

You should visit your nearest counseling center of AKTU.