When I say dogface puffer I mean the Arothron nigropunctatus which I believe top out at about 13"(in the wild). I would say for the most part liveaquaria is the authority on suggested tank size. They use to say 125g now I believe they recommend 150g.
My vote would definitely be for a sargassum trigger.
All the fish you are looking at are pretty hardy, I would say none are what you would call ick prone. Lionfish do not handle copper, even if they make it through they will likely die within a year probably due to organ failure. Eels are as resistant to things like ick and velvet as one can be, i would have no concern there. Internal parasites are rampant, so just make a plan to address that issue.
To consider about your eel decision, you really can't go wrong with a snowflake or zebra, the ribbon eel is high maintenance. Even as long as I've had mine, still a challenge to feed, and everyone I know is amazed I even have him alive after 1.5 years.
Line up sounds great. None of the fish are so overly aggressive that I would be concerned about the order of addition. I would just make sure you have the betta, lionfish, and eel well accustomed to their feeding ritual before adding them into the dt.
Dogface Puffers are also very thick, or chunky. So it's not just length, but width and height.
I think her size tank would be ok for a Dogface just because they don't swim much. They just slowly float around the tank, lol. But a fish that size, if kept long term, would definitely limit the amount of other fish. They do have great personalities to.
That's why I suggested the smaller puffers, though multiples of even different breeds will fight for a few days, unlike the bigger ones. I had 3 smaller puffers living in a 90 and every time I added one, one of the others would attack no matter how long the introduction period was. It only lasted 2-3 days, but something to watch out for.
Just my .02¢ though