My thoughts --
I think overhead light fixture is a good idea. To get them to germinate in the cold, you would need a steady bottom heat source. But it would still be better to try to get them germinated and growing to true leaves in the house first with bottom heat and at least 50Â°F ambient temp -- maybe basement?
In the shed, if you could create an enclosed/tented mini environment, there won't be as much temperature loss -- but you probably have better idea of how that works. I found it makes a huge difference in my unheated garage to at least put some kind of cover across the front of the shelves (mine are set up against the insulated wall adjoining the house -- no windows) vs. not. And I tented the top with plastic sheeting draped over the light fixtures during the night while the lights were off when ambient garage temps plummeted below 28Â°F (with the entire set up covered with double layer of plastic, young tomato seedlings survived with bottom heat on through the night... did not without). If you set it up to create a thermal (up) draft across the seedlings, you can get away with relatively closely covered set up because of the air movement.
Presumably, when sun is shining in the window it can get pretty warm in the shed even without heat? As much as 50's-60's? If 40's the seedlings will not grow much and will turn purplish, 50's they will grow slowly but will develop sturdy stems with good light.
If you are not uppotting, only fill the cups 1/2 full to germinate then bury the stems (not lower leaves) as they grow (kind of like potato shoots). But put the light tubes directly above the cups when they are not getting sunlight. This works better with clear cups or cut off soda bottles because they get more light. Personally I like recycling the soda/clear bottles better because of the straight sides.
I like screening the bagged mix with 1/2" hole riddle and cutting the sifted bagged mix with some coir and perlite or sand for seed starting, then uppotting to bagged mix straight or mixed with 1/4 sand/perlite depending on how heavy the mix is. In addition to drainage holes, I have been cutting 3-4 vertical slits in the side of the cut off soda bottles and tall drinking cups with an utility knife for better root aeration.
...hope this works out for you because it's a lot of fun to grow different varieties of tomatoes!
...not surprisingly -- Yes I have tried keeping seedlings together. Typically one will overwhelm the other. Once planted outside, IF there is plentiful nutrients and water, they can find access to what they need, but if there is limited resources, again, one will bully the other and I have observed overwhelming difference in development to the point where the second one might as well be culled....