B.Tech

You just need to score more than 50% marks in class 12th or in Diploma and score a particular rank in JEE or any other engineering entrance exam, then you get admission in any engineering colleges according to your marks.

Hello, B.Tech Computer Science is more preferred by student in comparison with mechanical engineer. As now a days, IT sector is increasing their range that is why computer engineers are more required. But, I recommend you to think about your interest instead of trend. And, if you are really interest

Yes, of course, you can get lateral entry, 90% of the colleges provided lateral entry. If you are eligible, you will get admission in B.Tech for sure. If you want to check colleges that provides lateral entry in your location, visit:
Shiksha.com

If you have got 50% in PCMB (Physics, Chemistry, Mathematics & Biology), only then you can get admission in Poultry Production or any branch of B.Tech in Tamil Nadu.

Yes, you are eligible to take admission in many institutions. The cut-off for M.Sc. in Mathematics is 50% aggregate. But in some colleges, you have to take an entrence exam. You can check the best colleges near to your location here:
Shiksha. Com.

You can apply for it. They conduct their own exam, you need to do well in it to get a seat there.

Yes, sure you can do your M.Tech from a much better college than your previous one. For that, you can give some entrance test to get the best college for you. You can give GATE for taking admissions in top IITs, NITs and IIITs. Best of luck.