Hanging a garden hose holder and need information on framing under a window

I think you should seal the wood for two main reasons:

1. the posts are in a bathroom, where they'll be subject to high humidity (possibly condensation) and temperature changes. That's a recipe for warping and splitting in any wood. Even with the thick posts used in a post-and-beam house, you'll still see some ill effects eventually.
2. Doug fir is rot-resistant, but not rot-proof. If you do get condensation in that bathroom, you'll shorten the life of those beams if you don't protect them.

A third (cosmetic) reason is that untreated wood is going to lose color over time, becoming gray. I would recommend you think about treating all the exposed posts and beams, not just in that bathroom.

This is a bad idea to try and build your own I joists. Often the building codes will defer to the manufacturers specifications for engineered materials like I joists, engineered beams and also engineered OSB board. This is done because every manufacturer is different and because these manufacturers generally have actual accredited engineers that are certified and vetted who test these things in a methodical and proper way.

A small shed, on a residential property, is still considered a seperate structure, and as such is subject to your municipalities building codes, zoning requirements, by-laws and building permits. A review of your plan or an inspection likely will not pass and code enforcement may require you to tear down the structure if they feel it is unsafe. Inability to comply with code enforcement can result in loss of your occupancy permit for the residence.

Code aside, I think even using engineered I joists for studs wouldn't be the best idea. While I joists are okay for hanging drywall on a ceiling, a wall tends to have a lot more nails, screws and anchors run into it for everything from pictures, TV's hanging on lags, etc... With drywall, the screws will be run into the sides of the I joist where they will not damage the OSB in the middle. Running a nail or screw into the middle of an I joist greatly damages the integrity of the joist, and there would always be the possibility that somebody would attempt this and damage the stud. For a shed I am thinking about possibly trying to hang pegboard on the wall as an example.

Furthermore I can't even imagine how one can safely frame for a window or door with I joists. What is the proper way to nail the jack and king stud together? What if the wall is load bearing? Engineered I joists are tested when they are laid out on a span, not supporting load vertically.

Just use dimensional lumber, or purchase pre fabricated engineered I joists for supporting the load on a floor per the manufacturers specifications.