I think I can give you a few informations about relevant courses as I am in the same situation and did a lot of research about those courses. Here are a few courses you should keep an eye on (just Europe).
Design courses with a focus on engineering:
TU Delft - Integrated Product Design
Brunel University - Integrated Product Design
Royal College of Art - Innovation Design Engineering
Loughborough University - Design Innovation
And here are some Universities with similar courses I dont know the name of
Politecnico di Milano
KTH Stockholm
TU Eindhoven
Glasgow School of Art
TU Denmark
There are loads of other courses that would aim your wishes.
About the Portfolio. Just look at portfolios from others to get some inspiration. If you have a good product idea, try to depict it. You dont have to build a working prototype (if you do its even better), just draw sketches, build mockups, create some renderings and give your projects a structure and consistance. Show that you are able to think creative.
I also had to learn it and I just had four month time to build up my portfolio. At the end it worked out, as I got at least an interview invitation at most Universities I applied to.