building off of what renovator said, if you have load bearing walls on the first floor, you need to align structure below them in the basement. This can be beams or walls. Frame walls will be your cheapest option in most cases. However if you do not want a wall there, then price is not always the driving factor. However if you are a single story plus a basement, you shouldn't have too many first floor interior bearing walls. If you use steel or LVL beams, they must be sized to carry the first floor load as well as the loads placed on the first floor (2nd floor and possibly roof loads). As you can guess, this can make wood beams, even built up, quite deep. Steel might be your best bet here if you have longer spans between columns in your basement to keep head room heights at ideal levels. I tend to disagree with the claim in wood costing more then steel. Steel will cost more then wood in most cases. The post can be hidden inside walls (why would you do this? save the money and make the wall bearing! unless of course you have a very high point load) or they can be boxed out and made decorative. I joists should actually be cheaper then a floor truss, but typically cost a little more then solid 2x8-12 framing. However IMO, I joists are far superior. They can be had in very long spans so they can lay a continuous 30'+ joist across your load bearing beams/walls and allow for straighter and faster construction. Also, depending on the depth of them and the spacing, they can easily clear span 20'+. Plus they are a "greener" product. Trusses will cost more, but the benefit you have there is the web openings to allow easy smaller duct runs and electrical access.

I would put hardwood floors throughout the area but I am partial to hardwood and know it’s not budget friendly. You could use rugs to create distinct areas — one for dining and then one with the couch you float. Then choose a curtain to coordinate with rugs. Then pull a couch color from the rug (I believe in choosing your favorite color for a couch unless doing a white slipcover which actually has been my best couch with kids because BLEACH haha. Most couches today just don’t last forever anyways — get something you will love). Then choose an accent colors for toss pillows, get an end table and some lamps, and a chair in your accent color. Art. Then paint. And maybe think about painting the brick and wait until the other elements come together to see how you feel about it. I think you would feel a lot better about the space if it looks more put together. Then for your storage, you could get ikea sektion cabinets or something like that and even paint them to have more attractive storage and there are even hacks to make them look more built in. I’ve hacked several ikea things in my house (closets, playroom storage, a bookcase.. and I am not a DIY person at all LOL but custom storage is so expensive and if it’s not in areas where I entertain or my kids are just going to beat it up, no way am I doing anything custom) you could maybe even add “mudroom cabinets” because your garage door leads into the dining room correct? You could totally hack ikea pax into mudroom cabinets if on a tight budget/. You could maybe also add a pantry cabinet if you need extra storage (I don’t think I saw a pantry in your drawing and this is where getting the full floor detail is really helpful!!) if you can better think about what you need, you can probably get more advice about what to do with that area.

I would not even trust a contractor get an actual structural engineer to check this out. I had a 1905 house and never assumed anything untl it was checked. by an engineer,My guess is the house is balloon framed so it is not as easy as just framing in a new header One of the tricks with balloon framing is you can't just support a floor, you have to support the wall above where you want to install a header. Screwing a 2x to the studs above the opening is the usual and you have a roof outside with a ledger. You might add more screws to that ledger and support it with something temp while you work on the door.

It can be, your best bet is to contact a local architect and provide you with the drawings so you can obtain a building permit.