Yes, it is possible. Yes, it is expensive.
Most fabs will need the design on a mask, to get the data onto a mask you should provide it in an industry standard of GDSII or OASIS.
TSMC will probably make a multi die wafer with your design being only one of ~200-4000 dies on the whole wafer. It will be manufactured with similar devices, sorted by die size and process.
The software is out there, most is expensive, but there are other options. Klayout will allow you to draw the patterns, but testing the structures and making sure the device does what you want would be a huge test suite of software.
The point being if you already know what to do for the design, using off the shelf freeware is ok, but there are a lot of complexities (hence huge corporations) to get everything right. The fab can help but they will charge you a lot.
As far as courses- digital design, analog signal processing, device physics, signals and systems. Those are the starting point. After those courses you will have fairly good base knowledge to expand it to the exact things you want to know.