I grew my cucumbers last year in 5 gallon buckets. That's probably more soil than they really needed.
I'd say you could have all these outside any time now, unless you're at significantly higher altitude than we are here, and have a lot later a last frost date. Our last frost date is around May 17th.
I think you might be creating yourself more work than necessary with things like the potatoes if you're starting them inside. Transplanting takes a lot of time/effort. I planted my seed potatoes outside in the garden practically as soon as the ground was thawed enough to dig. They just sprouted up out of the ground aa couple weeks ago. No transplanting necessary.
A lot of people around here also just plant cucumbers straight outside, albeit later than potatoes. They are a bit more cold sensitive. You could probably just plant cucumber seeds outside now and they'd be growing like crazy in no time.
As for soil, the cucumbers seem to appreciate a good soil with organics that'll hold water well and such. Cucumbers like a lot of water. Also, you might want to pick up some Epsom salts for cucumbers. They like the magnesium on occasion with watering.
Potatoes on the other hand are pretty happy in poor soil, but it needs to be well drained. For mine, I mixed sand with some rather poor top soil that I got from a local landscaping company. The top soil they gave me had quite a bit of fine gravel in it, and then I added sand on top of that. The potatoes do fine in that. I had a problem the first year where I gave them a very rich, dense soil mixture. The soil held too much water, and a number of the potatoes rotted. (I was also probably watering them too much, I am good at loving my plants to death)
Also, I grow my potatoes in old car tires. You stack up the car tires and continue to bury the potato plants as they grow. Just google "Growing potatoes in tires" if you'd like to know more on that.