Design Engineering

You can probably pursue mechanical engineering and alongside you can pursue a machine design course.

Yes, you can. Just aim for the relevant exam.

You can go for Canada. Colleges there provide the course you are looking for. The maximum cost involved would be around 14 lacs and you can work part time too for 20 hours/week. And they allow you to stay after your studies for the same duration as your course to look for jobs. If you manage to get a job in that time, you can extend your VISA to work there.
I hope this helps you to some extent.
You can always take help of certain free counselors if you are in some doubt.
All the best.

Entry requirement:
- The candidate must have a 10 + 2 or intermediate in any discipline. Students in their final year of graduation can also apply.
- Above 50% of aggregate marks in qualifying examination (First class preferred)
- Fluency in English with strong communication skills
- Aptitude for design
The fee differs every year, so this you will get to know only at the time of admission.