By IRC code definition, a conditioned space is a "space within a building that is provided with heating and/or cooling equipment systems capable of maintaining, through design or heat loss/gain, 50F during the heating season and 85F during the cooling season, OR communicates directly with a conditioned space. For mechanical purposes, an area, room or space being heated or cooled by any equipment or appliance." What this means is that you can add any type of heat to the under space and condtioning it. Doesn't have to forced air at all. It can be steam pipes that radiate enough heat to maintain and heating season temp of 50F or above...A strip of electric heat, radiant baseboard or any other means. Likewise, the space can be conditioned in summer by running AC to it. Likewise, the space is considered condtioned if it opens to another portion of the basement that is already heated and cooled by design or default. Likewise, the space is considered conditioned if it simply naturally falls between the 50 to 85 range at all times...even without mechanical means. The key to #5 above is that you cover the floor with a vapor retarder....insulate the walls from the top of the rim joist to a point at least 2 feet below the frost depth...and then to condition the space by design or default.

I'm not getting into the vented vs sealed crawlspace pro & cons. but to answer OP's question, 3.5 to 4' is sufficient working room for spray foam install. closed cell has a stronger smell than open cell IMO. but the smell fades after a couple of days with either. if you are anyone in the family are sensitive then install should be done when ample time is allowed for any smells to dissipate. I'd start by investigating the recommended install (joists covered or exposed). I'd think that 4" would be ample at R-28, but I don't know your requirements in your area. just remember that you have to meet the code minimums. R-value of foam into inches of foam. not "performs like R-xx" or "average fill" same depth or MORE...not less everywhere. No Q or K values, R-values only. it is the installer that is the key figure in the install, not the salesman. make sure you are on page with the installer & be willing to go into the crawlspace & measure low spots before they leave for the day. it is worth your time. and yes dehumidifier in vented crawl is trying to dehumidify a portion of the great outdoors. best of luck.

I suspect that the vent screens were placed first, simply for ventilation, and then they were tempting/convenient access points over the years to bring in gas/electrical with the least amount of work. It looks to me like the large holes, like the one where the gas line enters, were simply sloppy work when the line was put through, rather than the work of a rat. There is nothing wrong with having these services come through wire screens like this, but it will be a lot of work to get these off, replace them, and have it all repaired and looking neat. For example, how would you intend to deal with the gas line that goes through the center of the screen, meaning...how would you get a new, intact piece of screen on without undoing the gas pipes? Here is my suggestion: leave the screens in place and buy some matching wire to make neat patches. You can simply cut the patching pieces so that there are stubs of wire around the edges, and then these can be bent over the existing wire with a flat blade screwdriver and/or some needle nose pliers to hold them in place. If you don't like the idea of small patches, get pieces of wire mesh that are as large as the entire openings and then fix them in place over the wire in the existing openings. When you're finished patching, get some Rustoleum brand spray paint to make everything blend in and to deal with the existing rust.

10 mil minimum, seams overlapped by 6 inches and taped - the entire length of any seam. I like Gorilla Tape. "Brought up" 18 inches+ and taped around any non-wood supports (imagine wrapping a tree trunk) and up onto the non-wood foundation walls). Lay it loosely - no reason for it to be "tight". No stakes - you don't want holes through the barrier. Cover the plastic with 4 inches of pea gravel to prevent tearing (when crawling on it), the gravel will change color if a breach is present. Color of the plastic doesn't matter. It's a labor intensive, knees and back aching job, so have it done correctly this time.