Is it possible to have a single output with multiple skins?

I think it would be worth submitting a feature request for the theme swap. You can request features and report issues here: https://tracker.adobe.com/

Post the item number in the discussion so people can easily vote if they want the same functionality.

Let's first deal with the help being embedded within your app. That is something I cannot help with other than to point out the help relies on browser functionality so it will depend on how well you can embed that. I would certainly be testing thoroughly.

Any recommendations on what Robohelp skins to use that would work well with light and dark themes?  

My site uses a minimally customised Orange skin and the Dark skin looks to be the same layout and functionality as Orange, just customised to be dark. I would suggest for ease of maintenance that you use the same CSS in both with just the changes you need for the dark theme.

 I have been using the Responsive HTML5 output so far.  Any thoughts on whether to use that or Frameless? The built in styles seem quite limited, and I'm wondering to what extent I can edit them?  

Very definitely go for frameless as that is where Adobe are focussing all development on features and designs. Responsive templates have minimal customisation built in although you can do more with a user.css file. Frameless templates have a lot of customisation options. They are not well documented anywhere as so many individal settings interact with others and it is the mix that gets what you want. As you are a developer, I think looking through them will not be too difficult for you. It's just a learning curve. 

Is it possible to output a single set of help pages with Robohelp, but switch among light and dark themed skins by having the App point to different skins (CSS files)?  

There is nothing built in for this. I have a feeling this has been asked and that no solution was found. RoboHelp is designed to reference all the CSS files referenced in a topic and the CSS for the template. No switching option.

One solution would be to create two outputs and allow the user to choose before opening the help or letting them switch but that would not close the window with the default output.

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Hope that helps and definitely ask about Deepl in a different thread. That said the main forum supporters don't have much knowledge of it. Hopefully someone who can help will come in. If not it's a trip to Adobe Support. See https://helpx.adobe.com/contact/enterprise-support.other.html#robohelp for your Adobe Support options. The email link tcssup@adobe.com is recommended as it reaches a team dedicated to Technical Communication Suite products including RoboHelp.

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My site www.grainge.org includes many free Authoring and RoboHelp resources that may be of help.