உடல் பயிற்சி செய்

Yes, 'Udarpayirchi' is the grammatically combined form (sandhi), while 'Udal payirchi' is the two-word version. Both are understood, but the combined form is more standard.

No, for mental exercise, just use 'payirchi' (practice) or 'manap-payirchi' (mind practice). 'Udal' specifically means physical body.

'Sei' is formal and standard Tamil. 'Pannu' is colloquial and very common in spoken Chennai Tamil. Both mean 'to do'.

You say 'Naan gym-ukku pogiren'. You don't necessarily have to use 'udarpayirchi' if the context is clear.

Yes, it is a type of physical exercise, but people usually specify 'Yoga sei' because it's a distinct discipline.

You say 'Udarpayirchi seidha piragu enakku kalaippaaga irukkiradhu'.

Yes, it is standard across all Tamil dialects, though some regional vocabulary for specific exercises might differ.

Yes, this is very common 'Tanglish'. Most people will understand you perfectly in urban areas.

There isn't a single word, but 'soamberiyaaga iru' (to be lazy) or 'velai seiyaamal iru' (to be without work/activity) are opposites in spirit.

Ask: 'Inru udarpayirchi seidhaaya?' (informal) or 'Inru udarpayirchi seidheergala?' (formal).

It is 'Udal' (body). The 'l' becomes 'r' only when it joins with 'payirchi' due to Tamil grammar rules.

Yes, it's a very common phrase taught in schools during Physical Education (P.E.) class.