Good questions. The size/girth of the trunk just depends on the size of tree you want to create. The thicker the trunk (generally) the taller the tree. Of course a thick trunk can also be used to create a short powerful "sumo" tree. It's all a matter of taste. It is wise to buy or collect a trunk that is as big as you are looking for so you can begin the fun stuff, branching. I don't hesitate to collect younger stuff and grow it out personally, but that's just my style. Or was my style... my collection is almost non existent currently. I downsized hard. I can talk about this stuff all day but until you really start to see how much you can shorten internodes and what species will and won't play nicely, it's just kind of wasted air. I don't know much about birch, except that they are beautiful and can make great trees. If I were going to use one for bonsai, I personally, would be looking for a taller slender trunk because that is how I see them in the mountains and forests. A carrot sized trunk would typically be used to create a smaller tree. maybe less than 12" but the trunk can always be planted into the ground and grown larger.
As far as fall collecting, I have only done it a few times, but with a high success rate. But it was after a decent amount of time spent caring for potted trees. The problem is that no wounds will close over the winter so the tree sits open to fungal infections all winter. I don't know my betulas, but I believe they are a short lived species that are generally prone to that stuff. But like I said.. learn by doing. might as well try. I fall collected a a hop hornbeam a few Novembers ago. everything I read afterward made me believe they were fickle and hard to collect. I assumed it wouldn't make but it turned out to be one of the most robust trees I had every played with. Just sold it actually. Nothing trumps experience.