B.Tech

It depends on your JEE Mains and advance score, and also from where you do your B.Tech, but let me give you an advice please be more specific, while you ask questions, because it becomes very difficult to give correct answers. And, wrong answer will do wrong to you only. I'll suggest you to go for m

Admission in PSG College of Technology for BE in robotics and automation engineering is through Tamil Nadu Engineering Admissions (TNEA) counselling guidelines based on marks obtained in class 12th final examination.
For more details, check the link below:
https://www.shiksha.com/b-tech/course/b-e-in

Some Short-term courses related to game programming which can be done during your spare time are mentioned below:
1) Game Theory
2) Advanced Multimedia
3) Digital Media
4) Design of Games
5) Gamification Workshop
6) Game development and programming
7) Pixel art for video games
8) Introduction to photoshop
Re

This is very time and situation based question, nobody can guarantee that which company will come at which college for placements, because it depends whether the company has vacancy or not, if they have vacancy which college they will choose to go.

You can go through the ICAR exam or through other exams of the respective colleges and universities. Some good colleges are:
NIFTEM
OUAT
GBPAU
SHIATS.

About all private college give admission for direct lateral entry, but if your percentage is good in Diploma you can get good Engineering College. College which give direct admission are:
GNDEC Ludhiana
All PTU College
JSS Noida,
GL Bajaj,
Galgotias University.