You need to add a sketch of the room sizes and wall & window locations.

@tatts - not very helpful! We're not all good photographers, and maybe the lack of light is part of the design problem!

Not sure what your existing floors are, but if concrete, consider using concrete as your finished floor. Will be some expense for finishing and sealing the floors (depending on existing condition), but probably less than new floor covering. Concrete is an excellent finish floor below grade given the risk of dampness or flooding. There is a lot you can do, design-wise, as well, including colors and patterns. See the concrete floors in the photo attached. I love living with concrete floors, and enjoy adding thick or textural rugs for warmth and comfort. Let me know rough sizes of rooms, and I would be happy to give additional ideas (sounds like my own basement).

Not sure, but looks like u may have one large wall to work with. I would use that wall for the media area/built in's on the sides. There are a lot of really cool rooms with the tv in the middle of the wall/unit and side shelves/cabinets on bottom or variations of this. Look at the family room/basement pics on Houzz for ideas..u will find something cool! Also, I personally would go with carpet...easier to deal with an to change. Good Luck!!

First thing you want to do is vote for the guys/gals who believe in global warming.

idk how big these room are but u may have trouble fitting a pool table in the same room with a media ct. but if it's two rooms/like it sorta looks...may use one for table/other for t.v. etc. or hang tv on wall in pool room.

To get a pool table in there, you need about 20' at least. 8' table plus at least another 6' for each side so you can pull the stick back ( don't play, but ran downstairs to look at our table). We had to do what Jayme suggested and use two rooms. In the smaller we have a round 8' foot game table with 6 leather swivel chairs, a wall of cabinets with a bar fridge and tv above and wall to wall carpeting. Pool table room has porcelain floors over concrete.

We do concrete overlay systems on floors and trust me it is the best option for a basement floor. With radiant floor heating you won't have cold floors. There are many designs and your imagination is the key. Area rugs can add to the warmth too. Check out websites for what is out there.

Love the fireplace! Yes, the photos aren't grand, but we can all tell that it is a potentially lovely space. FIRST -- make the place as waterproof as possible. Forget carpet, engineered wood, and all of those options. Keep the concrete and finish as recommended above, or tile. In addition, with the concrete, or tile, use tile base trim around the floors. Use a rustic, natural tile to tie in with the fireplace. It will protect the drywall and eliminate most of the headaches of flooding. Lighten up with the paint. Make it neutral and let the natural stone colors of fireplace and tiles be the inspiration.

Hate to let a lake view go to waste; is the ground outside those windows at floor level? If yes, french doors in each section opening out onto a deck overlooking the lake would add a lot of light to the room and really open up the space.

I'm not sure what downspouts drainage problems have to do with global warming - I guess i'm missing that connection.
You've got some nice space with the fireplace as a great divider of function! If you're considering a pool table, just remember that a hanging light over the table is bound to get hit with few cues once in a while!