Is it permissible to combine the Dhuhr and Asr prayers and perform tayammum instead of wudu due to the difficulty of purification and to avoid missing the prayer time?

The basic principle is that each prayer should be performed within its prescribed time, and purification is done with water. There is no specific place required for performing ablution (wudu').

If performing ablution in the restroom is difficult, one can look for another place outside the restroom to perform it, or perform ablution at home, wear socks, and wipe over them at the institute.

If that is not possible, one can resort to the "apparent combining" (al-jam' al-suri), such as delaying the Dhuhr prayer until the end of its time and then praying Asr immediately after its time begins.

If this combining does not fit with class schedules, then one should go to the mosque to perform the prayer on time, even if it means missing some lessons, unless that would result in harm. In such a case, it is permissible to combine two prayers due to necessity.

As for tayammum (dry ablution), it is only permissible for one who cannot find water or is unable to use it.