First Question: Is 2" x 2" a common size and is it actually 2" x 2"? I
recall that a 2x4 isn't really 2" x 4".
Yes, 2x2s are common (at least in my area) but as you suspected, they are actually only 1.5"x1.5".
Second Question: Since the table will always be outside, what is a
good wood to use? Pressure treated or some type of hardwood?
It depends on whether your food and utensils will ever come in contact with the wood. For example, if you always put your food and utensils on a tray, you don't need to worry about the food safety of your lumber, and pressure-treated lumber would be fine. However, you could use any type of lumber and treat it with a UV-resistant finish, or you could use a naturally rot-resistant wood. Although species is not the only factor that determines rot resistance, it's a good place to start, as discussed in an article on garden.org:
- Exceptionally resistant: black locust, red mulberry, osage orange, and Pacific yew.
- Resistant or very resistant: old-growth bald cypress, catalpa, cedar (either eastern or western red cedar), black cherry, chestnut, junipers, honey locust, white oak, old-growth redwood, sassafras, and black walnut.
- Moderately Resistant: second-growth bald cypress, Douglas fir, eastern larch, western larch, old-growth eastern white pine, old-growth longleaf pine, old-growth slash pine, and second-growth redwood.
Third Question: Is a Mortise and Tenon joint the best joint to use? I
used one for the mock up and Tenon will have to be trimmed, is that
okay or should I change my dimensions.
There are numerous joints that would work in this application, from pocket joints as suggested by LeeG, to lap joints (including the fancier dovetailed lap joint suggested by Graphus), to simple butt joints reinforced with dowels or biscuits, to even the most exotic joints. You could even move the legs to the inside corners of the corner joints and use classic dovetails rather than making the legs integral to the corner joints.
Mortise and tenon joints would also be fine, but probably wouldn't work very well as pictured in your illustrations. The critical factor in determining a mortise and tenon joint's strength is the effective width of the tenon--the distance your router would travel when you cut the mortise (if you were to use a router).
If I were using mortise and tenon joinery on this project, personally I'd replace the horizontal parts with 1x4s, decreasing the thicknesses but increasing the widths of your tenons (and, consequently, your mortises). This will allow you to offset the mortises further toward the outside, in turn allowing you to cut slightly deeper mortises without any interference from the tenon that goes into the mortise on the adjacent face. As an added benefit, the wider and longer tenons will produce more long-grain glue surface.
If you want a purely mechanical joint and don't want to use glue at all, you could either drawbore your mortise and tenon joints or each corner joint could use a combination of an interlocking tenon with a tusk or wedged tenon. Just within the realm of mortise and tenon joints, the possibilities are endless--it's really all about your personal preference on this project.
But suppose you really love your design, and you want to use mortise and tenon joinery and all 2x2 material for the frame. In that case, you should use a double mortise and tenon to achieve more surface area for the joint. Marc Spagnuolo, aka The Wood Whisperer, has a great article and accompanying video on how to create a double mortise and tenon joint.