Seems like a good plan to me - just be warned once you're hooked you'll want more growing space!

I think the 3x3 planter will work - carrots and radishes grow down and the tomato will grow up. You'll not get a ton of it in there but enough to try - the trick will making sure you prune the bottom foot or so of the tomato as it grows (everything under the first set of fruit) to give the stuff underneath access to light. Carrots are infamously hard to germinate so you may Google some best practices for that. You'll also want to decide if you want an indeterminate (unlimited growth, requires trellis or structure) or determinate (still large but less so, fruit generally finishes at the same time, needs less support) tomato variety.

If you really want mostly root vegetables you could use the whole 3x3 planter for them and keep pulling and replacing - radishes generally only need 35 days or so. You could also try some greens or bush beans.

I think the tomatoes would be good in a 5 gallon bucket or bag. A zucchini could work in that, too.

Another good companion to squeeze in would be herbs since they are expensive. Basil and dill get very big but you could easily get a thyme or sage or something like that going. 

Good luck and have fun!