Polytechnic

Entrance exams will be conducted for Polytechnic (Diploma) courses within the states separately. Check the mentioned link for state wise latest entrance examinations:
https://www.govtjobexams.com/admissions/

You definitely can by taking classes at different timing. It depends on you how much stress you can take. Also, will your mind be able to capture and retain the different commands of the two software. Saying so, which of the two courses is better depends on which industry you go into next. In automobile industry, catia is used more, which in construction, autocad is used more as per my knowledge. All the best.

You are advised to appear in JEE Mains and JEE Advanced to seek admission in IITs after Diploma in engineering. IITs do not have lateral entry admission provision. DTU University conducts it's lateral entry admission test. Refer http://www.dtu.ac.in/Web/Admission/b.tech(LE).php.

You need to be a graduate to pursue MBA. Hence complete your graduation first.

Mechanical engineering is having more scope in the pvt. sector as well as government sector.