updated one photo in the OP. 07/26/15
I will use the good suggestions about plywood or at least more boards, to bind the legs together. I will also be using adjustable feet. I will have the T nuts and feet coming next week. They will have rubber soles.
As suggested, make a box to make it stiff: cover the back and sides, down to a few inches from the floor, with 18mm (or whatever weird thickness unit you use over there) plywood, screwed and glued with lots of glue between all surfaces touching. Two layers of plywood on top.
It really wouldn’t hurt to sandwich a strip of 1/2” plywood between 2 2x4’s and make the legs 3 2x’s with the strip. The strip of plywood will make 2 2x4’s the same thickness as one is wide, and will add tremendous structural rigidity. We always did that between 2x6’s for window and door headers building a house. 
I also think I’d bolt the legs on with at least 1/4” carriage bolts, instead of screwing them on. Overkill is underrated.
djozz is right about the plywood sides, and I’d see if I could find a 2 story house being built and talk to the foreman about getting a big enough scrap of 1 1/8” SturdiFloor for the top of the bench. That stuff is incredible! (Expensive to buy a whole sheet)
TiteBond is the glue you want, get the one with the green label as it’s more water resistant for your humid garage.
We’ve glued 3/4” thick Oak boards together at the edge and let em sit clamped overnight, then placed the joint on the edge of a work table and tried to break the bond. The wood itself would always break, about an inch away from the joint. Super strength when properly clamped, so if you used 2 sheets of 5/8” plywood glued with this stuff and screwed down tight, it’d probably serve as well as the SturdiFloor.
If you can find a second hand solid core door they make excellent table tops
Dang! How could I have forgotten that? A solid core door is 1 3/4” thick, at the size of this bench that’s dang near the equivalent of steel!
A laminated beam too, but it’s difficult to find a contractor using those with some scrap pieces cut off. Sometimes happens, but rarely.
What size is the bench Justin? Might help some of us that are more local to secure something ideal for the application. (Steel Plate, Aluminum Plate, Granite…)
Edit: I might have a solid core door I can cut off, if so, I can cut it with a SkilSaw and bring that up along with some stuff I was about to mail. Wouldn’t hurt me to take a drive in my new VW….
Looking good Justin, solid core doors are particle board inside, not particularly resilient. 2 layers of plywood or cross lapped planks is good. I prefer ply for it’s uniformity and continuous nature. Stiffest when spans run in the direction of the outer layers(more of the layers run in this grain direction). Glue everything. If it were me I’d be sick of all the advice but I guess it’s the string attached to a donation. 
I hear ya Scott, and you’re right. Justin has a plan and it’s being implemented. Enough said.
Solid core slab doors make great work tops. It is best to either laminate them or cover with 1/4 inch masonite.
The point about a solid core door is that while it makes an admirable work surface it isn’t designed to carry any weight or hold up to these kinds of stresses especially after cutting a hole in it. Only the outer rim(maybe 1 1/4-1 1/2” is actually wood). Most have cores that resist fire, sound, and warping but are not very strong. Some are made of foam. I can’t think of a single one of the many I’ve cut into that I’d recommend.
I know what kind of door you’re talking about, and honestly, I agree with you about those particular doors. However, if it happens to be a commercial grade solid core door like we see installed in commercial offices around here, those are usually solid particle board on the inside with a 1/8” veneer of furniture-grade wood on the outside. I wouldn’t doubt you could support the lathe very effectively with those doors.
Yeah the door you are talking about Rufus is known as a hollow core door. They typically have foam, honeycomb cardboard, gypsum for fire resistance, etc… inside. Solid core doors are completely different. Heavy strong and stable.
Particle board swells if it gets wet and crumbles if stressed. It bows under weight and warps if paint dries unevenly on either side. The only way it works in a door is if the door is an interior door and hung vertically so it won’t sag under it’s own weight. Try and bolt something to it and it will keep crushing until the washers meet. Unlike a solid wood door you cannot cut one down to fit a smaller opening without removing the styles or rails that allow hinges and strike plates purchase and keep the core material from crumbling and falling out. No offense intended but it’s snot, plain and simple, good for filler only.
One more photo in the OP. The base is all finished.
Looks really good!
The stand is coming along nicely! If you decide to not wrap the lower legs with plywood you could always run a 2x diagonally from top to bottom (if racking becomes an issue).
Putting a layer or two of plywood on the top is a good idea. I also like the diamond plate idea (or at least some thin layer of metal to make it easier to clean up the cuttings/shavings off the table).
If you were closer I would have limitless wood and top materials you could have used. I am a professional commercial trim carpenter by trade and always have stockpiles of material.
07/27/15 Updated the OP.
I’m a Carpenter/Contractor and know something about workbenches as I’ve also done my time as a Mechanic (before computers took over). First doors- you want not just a “solid core” door but a “Lumber core” door for maximum strength. Inside these are like butcher-block. As long as the base was solid and stable and kept dry, a solid core would do OK here. Sturdi-floor also makes great benchtops and Home Depot has a generic @ $13 something a sheet. Be sure to get the correct side up as only that side has rated water resistance. If solidly built, the entire base could be 2X4 but do add ‘racking’ support for the legs (like the plywood skirting). I’d prefer the top rails 2X6 or 2X8 just because that gives more to mount the legs to. Glue everything with polyurethane (Liquid Nails). Carriage bolt the legs. This will hold a smallblock V8 shortblock assembly without complaint so it will certainly hold the lathe! Once you have the lathe mounting bolt dimensions(or whatever your mounting), add rails underneath from front to back to attach to. Make these out of whatever you’re using for the rest of the top frame. Once you’re built set the bench in it’s permanent location and level it. If on a wooden floor, set the legs on 2X8 plates laid flat across the floor joists if you’re building engines- probably don’t need for the lathe. Add “L” brackets to the bottom of all legs, if on concrete “Tapcon” screws are a great fastener. If it’s against a wall, tie it to the studs with screws through the back upper framing from underneath. If it’s free-standing/portable or on wheels, you can increase the stability by having a shelf at the bottom and adding sandbags, bricks, scrapmetal, or whatever for ballast. If you’re bench is narrow enough you can add a laminate countertop section on it; waterproof and highly oil-resistant but can be a little slick.
Lathes and large bench tools were once all mounted on wood benches as metal was not easily available everywhere like it is now, so you can be certain that with a good design it’s enough for just about anything you’d want to do. For advice on this and other home related projects, I recommend you surf to This Old House Advice Forums where you’ll find a bunch of guys whose overall knowledge of everything “house” and “wood” is unmatched anywhere. I’m there as “Mastercarpentry” (shameless self-plug!).
Phil
Just a quick update. I still had not heard back from Grizzly, so I called them and they are just doing nothing, waiting for the trucking companies to e-mail replies to them, so maybe next week, if they find out something. They did not seem to eager. Dan warned me they were difficult to deal with and after the last call, I believe he's right. Hopefully, something will happen next week. I doubt anything will happen this week.
In the meantime, I am making a drop chute out of Metal Flashing. I will get some photos along the way.
07/30/15 Photo in the OP, of the catch pan mounted in the bench. Still waiting on something in the way of a shipping quote, from Grizzly.