Well, I usually plant mine out in June (here in southern Scotland), but I need to start them fairly early to have them big enough by then - yes about slim pencil thickness. However, if you need to get them in earlier because of their size or crowding, then you might have to sacrifice some depth. Make sure they are started off in good compost, then give liquid feeds every week to give good sturdy plants.
How deeply you can plant them in their holes depends partly on what your soil is like I think. If it's very crumbly then the holes may fill in quicker than the little leeks can grow, so they would be buried. I don't trim the leaves of mine so there's a chance something can peek out of the hole, but I think that might be wrong (see lower down).
For varieties, I usually grow a very winter hardy variety, having lost a whole crop in our first (extremely cold) winter here because I'd bought plants at the market and they must have been an autumn variety. I like big sturdy chunky leeks, not slender cordon bleu types, leeks which make lovely warming leek and potato soup in winter, or provide a good veg portion. I usually grow Musselburgh as being the old fashioned standard, but there are some recently developed varieties based on the type.
There seem to be leeks designed to be ready from early autumn to right through the winter, so you need to choose which you want - you don't have to stick to one variety. Which you choose will depend on your soil and local climate. I tend to try something new each year somewhere in the veg patch, so you could try out several varieties of leek and choose which you like best.
Grandfathers always grow wonderful leeks for some reason
I think it's to do with using 'old fashioned' growing techniques, which probably do involve trimming both the leaves and the roots, which I think will make each plant concentrate on developing a good, fresh root system which can head straight downwards (the existing roots may curl around in the hole and struggle to go down) before it starts putting on top growth, or even has to support what it already had at planting time.