I thought they were 2-2 x 4 with a piece of 1/2 " plywood in between. 
You can do it that way, but it’s more work.
It’s common to do that with any 1/2" material in the center. It’s matching the depth of a doubled 2x4 (3") to the width of a 2x4 side (3 1/2"). For a lintel or header, it’s the vertical 3 1/2" that counts most for strength, not the horizontal depth. Using two 2x4s is often easier than using a 4x4 because you can’t cut the 4x4 wood with a single cut on a common circular saw, and often no 4x4s are available while there’s always an excess of 2x4 lumber.
I see a lot of new construction, and Structural Plans always call for 4x headers. On a professional job, you plan in advance what material you need for things like headers, so that isn’t a problem. DIY, and all bets are off.
Seen it both ways. Depends on what’s on hand.
Sure if it’s in the materials list they’ll have the 4x4 material. But I’ve seen specifications for headers that only provide the depth of the header, as 2x4, 2x6, 2x8, etc., and the builder is expected to know how to construct a header. CA has some strict building codes and possibly using 4x4 is required there.
Since we’re having the OP rebuild his entire house now, I’d suggest checking the joist size on the floor under the fish tank. Besides reframing his windows he might have to rebuild the floor also 
Back to just the fish tank, I don’t think the wall strap is useful unless you have a weak or unstable stand to start with. Fish tanks are heavy, and would be very difficult to tip over. A sturdy stand is the best way to ensure safety.
I think everyone in the US is on the UBC now, just like CA.
If we’re going to talk floor joists - I would set the tank in the room so that it runs perpindicular and accross as many as possible - and if you have to have it running parallel - plan your stand to be atleast 18" deep to account.
While a 300 pound tank is ‘not much’ - if you don’t set it correctly, you may end up causing a weak floor structure by splintering/splitting the underlayment.
From my structural engineering days, that’s basically correct.
A structure can fail in several ways:
Compression - In the case of the OP’s fish tank frame, the weight of the tank would crush the vertical 2x2 columns, causing them to splinter. This is unlikely as the compression strength of a 2x2 of wood is well over 1000 lbs.
Buckling or torsion - While the cross section of your 2x2 can handle the compressive force, that is assuming an axial load directly applied along the center of gravity. In reality, slight imperfections and uneven load distributions actually result in a somewhat “eccentric” loading. In laymens terms, the longer your vertical columns, the more likely it might buckle or twist (torsion) under the same weight. IOW 6" long 2x2s will support a 300lb fish tank but 10 foot long 2x2s won’t. Not without some sort of cross bracing.
Shear failure - Unlike a buckling or torsion failure, your individual columns will remain intact. But a slight push to the side will collapse the whole thing like a house of cards. It doesn’t matter how think your columns are because the failure is in the joints.
My advice if you want to make a solid fish tank frame:
Take 2 38" x 14" pieces of plywood. (2" wider than the base dimensions of your tank).
Make a box using the plywood sheets as the base, top and 4 2x4 s the desired height. Use appropriately sized nails and wood glue for each connection. The long side of the 2x4s should be parallel with the long side of the tank
Cover the back and sides with sheets of plywood, glueing and nailing all along the edges.
Nail a 1"x1" edge around the top that the tank will fit into.
That should be pretty solid. 2x2s would probably do the trick, but why take chances for a few bucks? Also, a 2x4 gives you more space to use heavier nails.
I live in a ground-floor apartment with no basement. So I guess the floor is likely cement under the carpet?
Oh, and also- this stand is being built in my boyfriend’s basement, and will have to be transported here, and it’ll also move with me one day when I move, so if it’s structurally possible that it can be built safely/sturdily with 2x2s, then i suppose it’ll be that much less difficult to transport.
The commercial ones I have assembled are made of 3/4" plywood or particle board, and I’d hate to move one. The racking strength would be shot to hell, that is, they would become wobbly after being push and shoved around.
If it is to be moved, I’d go for 2x4s and screws.
To give you an idea of how strong wood is, here is a video of a “toothpick bridge” contest, where a bridge made only out of toothpicks and glue, and weighing no more than 100 grams ends up supporting 3,000x it’s weight (!).
Depends on if it is built on a slab or if it has a crawlspace.
Maybe. There’s almost certainly a cement foundation, but there may be another layer between you and the foundation to accommodate electrical conduits or plumbing. It’s hard to say.
hmmm, I don’t like the sound of this. My BF lives an hour from here, and intends to transport this in the back of his truck.
Well, you can listen to the people who are telling you what you want to hear and build it out of 2x2s, or you can listen to the people who know what their talking about and build it out of 2x4s. 2x2s spanning 3 ft and at a counter height of 3 ft will be wobbly as hell unless you brace the shit out of it. In which case you might as well build it out of 2x4s to begin with. And definitely use screws, not nails.
You can do something fancy like taper the 2x4s at the bottom so they don’t look so bulky or just drape the whole thing in fabric to hide the structure (the easy thing to do). Framing lumber, Doug Fir, is not very attractive anyway.
The plan is definitely to skin it with a decent quality, stain-able 3/4" plywood, so whatever frame there is would be anchored by the skin (and screws, not nails). According to my woodworking boyfriend (y’all are my second opinion 
), a 2 x 2 frame with 3/4" plywood is way more support than the store stands provide.
Weight wise (probably cost wise too) I think you would be better off with 2 by 4 s. Use 1/4 or perhaps 3/8 inch plywood for the sides and back. Use good long screws, along with good quality wood glue where any wood surfaces come together and that sucker will be plenty rigid IMO. Give the wood glue as much time to dry as you can stand before you load it up with the tank.
Now I am confused. If you are definitely to skin it with decent stain-able plywood then why not consider Shop steel metal shelving at Lowe’s? There are good explanations, perhaps no Lowe’s with reasonable distance. The question is part rhetorical and part inquiry.