In which mode does the IIITH UGEE application form available?

No, it's basically not possible.

The 5-year dual degree through UGEE (like at IIIT Hyderabad) has a completely separate admission process. If you join through JEE Main (B.Tech), you can't later switch into the UGEE dual-degree program.

No, despite being a "Research Aptitude" test, the computer-based format only supports objective types (MCQs) and numerical entries. There are no descriptive based questions in the IIITH UGEE exam.

The SUPR section is comparable to JEE Main, but REAP section is considered challenging as it requires creative thinking and "out-of-box" problem solving. It can be compared to the Olympiad level questions.

No, physical or on-screen calculators are strictly prohibited in both sections. 

Yes, in REAP section there can be questions on linguistics puzzles testing pattern recognition in unfamiliar languages. These questions do not require prior knowledge of the language, only logical deduction skills.