In Salesforce where can you see what edition you have?
In Classic, your browser's title bar usually tells you, e.g. Salesforce - Enterprise Edition. Alternatively, you can go to Company Information in Setup, and you'll see the Organization Edition field.
In Salesforce where can you see if the API is enabled? Is it on a per install or per user basis?
Both. The org must have it in order for users to use the API at all, and each user must also have API Enabled in order to use the API for their account. I don't believe there's an easy place to tell if your org has API access, but you can click on Help to contact support; they can tell you definitively.
Which edition do you get during your free trial?
You can get a free trial for almost any edition these days, but you need to find the correct form. Depending on the link you used, you might have any of the current offerings that salesforce has. If you want to sign up for a specific trial edition, but can't find the link, contact Sales.