I would space them more like 15-20' apart, maybe even more - don't forget that these clumps are each going to get 40-60' tall, maybe more if they live a long life, with a total width for the clumps of anywhere from 30-40' each. While you seem to want the crowns to grow into each other, you probably want them to fill out more than they would if they are really close together and shading each other.
I would plant nothing else in the bed for the first year - that way, the birches can get established, and not compete with other plants' roots. If you can't bear the thought of planting nothing, I would go with annuals for the at least the first summer, as they are fairly shallow rooted, and won't compete. Whether shade annuals or a mix for sun and shade won't matter too much the first year, as the amount of shade will be minimal. And if you can't stand that idea, then most of the shade perennials, especially those listed for dry shade, should do, although, as said, the amount of shade the first few years will be a lot less than say 5 years along. My brother has ferns, hellebores, mahonias, azaleas and a few other plants I don't now remember under his river birches, now 15 years old, and probably 25-30' tall, with sunnier annuals around the edges. The beds get watered pretty well, though.
Mulch well, about 4" deep of an aged hardwood mulch, no mulch in the 3-4 inches next to the trunks, and extended out from the trunks at least 4', 10-12' would be better. In fact, you/your husband could make the large bed linking the clumps with mulch, using cardboard or about 6 sheets of newspaper under the mulch to smother the grass, and not have to dig as much. You can plant through the mulch/paper even in the summer with annuals, and by fall, if you water as needed throughout the summer, the paper will have broken down, and you can plant whatever perennials you want through the mulch with no problem.
Do water at least an inch a week, for the first year, and every few weeks in the second year, if the rains get sparse. And if the nursery crew plants the trees, make SURE that they don't plant them too deeply - the flare where the trunk and the roots meet should be at ground level, not buried. If you can't see the flare when the trees arrive, brush away the dirt until you are aware of where "ground level" should be. I am assuming these are Balled & Burlapped trees - if they are in large pots, the same thing applies. If B & B, make SURE that the crew removes as much of the wrappings as possible, including any rope, wire or the burlap if it is a synthetic, so the roots can grow freely. And if potted, make sure no roots are running around in circles around the root ball(s), as this can lead to girdling and poor growth in future years.
Hope this isn't too much info, nor info you already know, in which case, my apologies!
Q