Science

Yes, it's definitely valid. We accept the degrees from the UK and there won't be any obstacle in your job endeavours.
Best regards.

Yes there are colleges offering integrated B.sc and B.ed courses - Sharda University, Indian Institute of Teacher Institute, Regional Institute of Education, Bhopal/Bhubaneswar
Lovely Professional University
Chandigarh University
G.D.Goenka University
Arunachal University

The cutoffs for the year 2017 was:-
1. General - 97.5.
2. OBC - 94.5.
3. SC-91.25.
4. ST-81.
5. PWD-86.25.
6. KM-87.5

Tezpur University is really a good university and has earned the recognition from all over the country. To crack TUEE, you just need to get your NCERT basics clear and sorted. After that practise some questions from the previous year papers of Assam CEE and you are good to go. The difficulty level of the paper is not much, it's the limited number of seats that offer massive competition. As for the 2nd part of the question, there is IIT Guwahati and NIT Silchar which can offer you your recommended course but I am not sure whether it is an integrated one or not. Hope this helped!