How can I build a screen door?
Personally, I'd get the table saw a portable folding stand, you can do basically everything you need with the tablesaw and can build any of the jigs you need. 

You could also do most of the work with your circ saw and router but getting the wood to the correct dimensions is going to be easier with a table saw since you don't have a planer or jointer. 

Instead of mortise and tenon, look at doing half laps for the corner joints and for a support brace or two across the middle. You can probably get some nice looking metal braces to strengthen up the corners as well.  You should be able to do half laps using your circ saw or router. You can also profile all the edges with the router to making look nice.