OPE engines will vent exh gases via blowby through the vents, it still needs good ventilation. The eu2000i models also move a LOT of cooling air.... very powerful fan section in that thing.
1- pipe the exh to the outdoors. Fabricating that could be a challenge, the nipple on the muffler output doesn't give you much to work with. May have to pull that and go back to the cylinder head if you want a solid connection.
2. make sure the pipe pass-through keeps it at a distance from flammable surfaces.
3. I would monitor summer temps during loaded tests in the summer to make sure the intake air stays cool. If you fan force, you will want the volumes and certain pressures that come from centrifugal fans---- a window fan isn't going to cut it. A gable fan would work.
4. the 12v output is rectified but not regulated. it will drive a small-medium DC fan ok. IIRC it is also not rectified full wave. an electronics fan would be useless both in terms of flow and compatibility with the dirty DC power. A radiator fan will pull waaaay too much current. A cabin A/C fan would probably be ok. you'd want to check it with an ammeter to make sure you don't kill the windings on the gennie. The 12V output WILL increase/decrease with engine speed. It will be lower than 12V in eco mode until it revs up. cheaper to use a 120v fan.
IMO a fenced square with a roof for protection from the rain would be a much easier solution. OR -- a protected "shelf" external to the building itself. You could shield it from rain, and either chain it (case is plastic...
) or drill a security bracket thru the bottom, or even install a removable cover for the shelf. BUT--- I might worry about gas-fed OPE sitting so close to a wooden structure. make sure nearby wood is shielded from potential flame-outs from the muffler, fuel spills, etc..
nice generator btw. very good machine. consider an external run tank for longer runs, it handles them well.