You could put a door in that greenhouse. Actually, you say you are considering taking it down? There is probably a door in there to get into the greenhouse that would become a door to the back yard as soon as the glass wall is gone. That big well frightens me. What happens to it after it rains? Is there a drain in it? Any sign of basement flooding? I'd arrange a visit during a downpour and look for signs of flooding...

We would have to consult a contractor and designer for other parts of the house too (new floors and kitchen remodel) so we would get quotes for the back room removal too. But we are stumped on how to have indoor/outdoor living back there. Replace windows with sliders?

FWIW it is worth to those looking at the options like Localeater. Did some research today. A 10x14 pergola starts at about 4k (pine) and goes up based on material and features. We like Walpole (have bought from them before) and they make a nice one in Azek (looks like wood but wears better ... we used it for new steps at our beachhouse) for 10k. The retractable sun shades for the top are another 2k. 8 weeks delivery, alas. Then of course we need the mosquito curtains, another 1k or so. We would also need to dig up our stone patio to put in a foundation, then the put the patio back and drill through it. Meh. An awning with Sunbrella is about $3200 for a 9x16 (that size would nestle in nicely near our koi pond area). 6 weeks plus. More expensive if it is retractable, but that kind cannot accommodate mosquito nets. The awning vendor had a product that was like a screen that came down at a push of a button, hidden up under the awning. That was another $10k though. Alternatively, you can use mosquito curtains. Not crazy about any of these options so far ... we might start simple with mosquito curtains for our french doors off the MBR in Maine and see how the y work. I like that we can get them in white...

I don't know where you live, but really think hard about what you want to do outside. Where I live, our time outside is limited to before "M" hour (mosquito) and never during big mosquito hatch-outs. So not that often. For hanging out in the warmer months, we use our screened in porch. The three things we do outside occasionally are have a campfire, grill, and I have a lounge chair in the front where one of us can sit and watch the other putter in the front yard. At my last house, I did invest in a staged redo of my back yard, and I never regretted it. The hours and hours of pleasure I spent out there added up fast. But we didn't have mosquitoes at the old place so were outside for dinner many, many nights. Plus, I designed a pleasure garden for viewing from an outdoor swing, and we often just sat out there at night and watched the garden. It had lots of white flowers that looked good at night, and lights and a fountain. The reason I am saying all this is if you are going to spend the money, really think about what you are likely to do out there. Maybe you need an expanded outdoor cooking space? If so, I'd be more likely to create an additional screened in area. Also, if you want to sit out more, like around a campfire or something like that, plan your landscaping around the views you want from the seating areas. Finding a good landscape designer is not easy. Mine was pretty bad. I could have done better myself and just taken some ideas to the installation contractors, which is what the designer did anyway.