Degree

No, it is not possible for you to appear in any TET without B.Ed or any other diploma. Either one or both of them are compulsory for TET.

After joining the job, you won't be able to continue with the regular course but if you are in the final year then you can contact your department and take a late joining in your department if they consider your case.

Follow the link below and filter colleges according to fees:
https://www.shiksha.com/it-software/colleges/bca-colleges-india?uaf[]=location&ts=20780472&rf=filters&ct[]=130&bc[]=33&ed[]=et_20

Hello! https://iimcat.ac.in/per/g01/pub/756/ASM/WebPortal/1/index.html?756@1@1#!
Kindly visit this link and check the "registration" section under FAQs about the eligibility criteria of the CAT exam.

Since ICWA degree is equivalent to graduation, most probably you are eligible to take admission. But still, you can confirm it by contacting the institute or physically visit the institute.
If you are a working person and want to enhance job value, position in your organisation and in your career, surely it is appropriate.