Window A/C into Wall A/C space?

In short, yes, you can use a window unit in place of a wall unit in that space... but:

Window vs. Wall units are more a matter of BTU than anything else. Lower BTU units will be window units, then in a range of "medium" BTU you'll find both window and wall units, and in higher BTU you'll find they're wall units... and some quite big, like you see in hotels/motels.

Don't think of window vs. wall unit. Instead shop for the BTU you need for your area to be cooled, and go from there.

Most window AC's condenser air intake are on the sides, install with caution

I think that with all a/c's we've had in that metal case that goes through our wall, except the 1st one (that was made to slide almost exactly in that metal sleeve) we've had the intake on the sides. They all have lasted about 5years, maybe the life of a window a/c? We stretch the window 'curtains" attached to the unit sideways, then put a piece of wood or styrofoam between the top of the unit & the roof of the cage...then put 2 layers of insulation tape all around the edges. Since the window a/c is smaller than the cage by about 2-4 inches on each side, I guess the intake works...but not as well as it would if the vents were on the back.

Aha...maybe that's the difference between a window unit (10,000 btu's for $200-$300) and the wall unit (10,000 btu's for $300-500)? Maybe wall-units have the intake vents on the back...! Also, I understand the wall units come with a new "cage" as they assume it's to go into a brand new hole... That's then one reason they cost more... We'd rather just put one in our cage...

But, otherwise, you feel a 10,000 or 12,000 BTU window unit would be fine, if it fits in our wall-cage & the intakes are left uncovered... maybe toward the back of the two sides, so they are actually outdoors, not walled in?

BTW, our unit goes in a room with NW exterior walls in the Phoenix area, in 100-115 degree heat for 3 mo/yr. The room is 15x18x8, so 270 sq ft. with one 3'x4' window & 3 doors: an outside door, one to the kitchen and one to an evap-cooled garage, all with well-fitting doors, which we leave shut. We think there is a small crawl-space under the roof with insulation over the ceiling, and that the room is built on a concrete slab.

In one suggested scenario, which did not take heat into the calculations, we came out needing a 7-8000 BTU unit. In 2 other calculation setups, which included the heat & window & NW exposure, we came out needing a 9500, then an 11500 (thus 12000) BTU unit. What would you suggest? Our current unit is a 10,000 BTU window unit. One installer said (phone chat) that he'd buy more btu's than needed, as it's so hot here. Another (and several web articles) said NOT to get more btu's than needed, as it will be clammy... more expensive... & the unit will cycle more & wear out faster. Gad. A new a/c sounds like a simple deal... but what to GET?
TIA for any suggestions...