Jaws not reading screen text until I click on slide

Best way to publish - html5 or swf?

I will recommend you to go with HTML5 since most of latest browsers are no longer supporting swf.

Best browser to use?

As long as you will choose to go with HTML5 then you have nothing to worry,your contents will play in almost all popular known browsers available in the market.

Best way to test - webserver? LMS? locally? preview in web browser?

If you are looking on testing SCORM/xAPI/tinyCAN then you will need LMS. You can test in local machine although webserver is always recommended.

Current issue is that jaws will not read my text captions on my slide until I click on the slide. Shouldn't it automatically read the text captions? Especially if there are no interactive objects on a slide?

I will recommend you to make it auto display unless you have interaction in it for that purpose.

Hope this helps!