> Can I start a company?

Depends on where you are from exactly.

If you are an EU national, then you can. Otherwise you will first need to get a work permit (against a promise to start a company, reviewed and approved by the canton) and it's generally a slow and expensive hassle requiring an experienced attorney. Whether this will make your company eligible for a startup loan is an open question, but I'm pretty sure that it will.

If you have a Resident Permit (type B or C), you can start a company. That means you should already be in switzerland with a regular employment contract (or as a spouse of a swiss citizen or swiss resident). What I do not know is if you can ask for a permit in order to then start your company (i.e. come to Switzerland for the specific purpose of starting a company), although apparently Swiss authorities are very helpful if you choose to move your already existing company to Switzerland.

A "Swiss Startup" is a startup actually based in Switzerland. As far as I know, foreigners with permanent residence in Switzerland have the same rights of Swiss people, plus more than 25% of population is actual foreigner in Switzerland, so that should not a problem at all.