Hi Autumn, 

Thanks for reaching out to the Extension Master Gardener Helpline. Great question!! So, as you know, stratification is the process of simulating natural conditions that seeds experience in the soil over the winter months to help them break dormancy and initiate germination. It improves germination rates and can speed up the timing of germination. 

It is getting a bit late now as, on average, seed needs about 10 weeks of cold (under 40 degrees) and damp conditions for stratification success. But it is worth a try as these actions could improve the germination rate at this point (note you can certainly plant unstratified seed but expect lower germination rates). 

The tote with sterile media should be fine -- you will want it to be moist but not soaked. Make sure to place the tote on the north side of a fence, tree line or building to keep the seed cooler for longer. You will also need to protect them from mice, voles and birds who might consider them a tasty treat.

I would not store inside your shed -- too warm in there, even if it is unheated.

You may get some germination but that does happen sometimes -- if so, at least you know that dormancy has been broken. I'm assuming the seed is a mix of multiple species so you can expect differences in germination as each has its own preferred timing and temperature. 

Good luck!!