Engineered wood or LVT.

Don't know about anything other than Carpet that you have listed.

I got a quote of $2200 for 400SF for Shaw Find you Comfort 2 - out of that, installation was a base charge of $500 (if you are doing larger areas, it is $0.99/SF with free removal of existing carpet. If you had existing hardwood, it is about $1.50/SF for demolition)

Carpet pad depends on what you choose - could be anywhere between $0.50 to $1.15, a decent 8lb carpet would be $0.99/SF. Carpet again depends on what you choose.

You get good rates online from dealers like Georgia Carpets. If you are wary of online ordering and hiring someone from thumbtack to install, you can get a quote from them and could try having a local company to price match or at least close enough to their material quote including shipping.

If you want to go with tile, check out Floor and Decor in store and ask them what their installation costs are. Also Lowes for Carpet. They sometimes have promos. We got it through a flooring company though.

Need to figure out flooring for one approximately 250 square foot room (newly constructed addition.) What are our flooring options, costs, pros and cons?

Things we are considering include:

Just using the concrete slab (polished, I guess) without putting anything on top of it
Carpet
Tile
Marmoleum
Cork

Would prefer not to do LVP/LVT but will consider it if nothing else makes sense/everything else is way more expensive. (We'd like to minimize toxic chemicals and environmental impact, but aren't willing to pay more than a few thousand extra for that.)

Any other options we're missing? What are the installation costs for these different options, and do any of them have particularly high fixed installation costs (i.e. don't make sense for just one small room)? What do you like and dislike about your flooring?