Developer a web browser with Adobe AIR

To answer your first question, yes* you can using the HTMLLoader class in AIR. Having said that, welcome to the great expectation.

*AIR has a 7 or 8 year old version of WebKit built in that the HTMLLoader class uses for rendering web pages. Because of this, some HTML/CSS/JS may not render or behave as expected based on experience with current web browsers.The AIR community has been officially waiting 2 years for Adobe to either update the WebKit version built in or provide a different solution to HTMLLoader for desktop AIR apps.

To answer your second question, I believe so**.

**You will have to use AIR21 for forever and all time. Based on the release notes for AIR22, it states, "Starting in AIR 22, applications that play swf content via the HTML control (WebKit) will now load the system level NPAPI Flash Player provided by Adobe (https://get.adobe.com/flashplayer).  If this plugin is not available on the system, the end user will be prompted to download and install the plugin from Adobe.". So that implies to be that AIR has bundled the Flash Player with AIR since AIR was first released and with the AIR22 release, that is no longer the case. So, if I am understanding that note correctly, you should be able to deploy an AIR app using the HTMLLoader class to create a web browser that can also load SWF files as long as those SWF files were created targeting Flash Player 21 or lower.