Definitely remove the exposed roots, and, if it's practical to do so, leave the fresh cuts exposed for a little while. If you cover them with dirt, there is a little bit more chance of pathogens entering through those cuts. This is a spectacular time to prune the roots that need to be pruned. Doing it now will minimize the time it takes the tree to seal back over the cuts. I can't tell much from the picture, especially since it is appearing upside down. I tried looking at it by turning my head as much as I could without suspending myself from the ceiling, but that didn't really help. From your written descriptions though, it sounds like you are right on track.

The mushroom compost would be fine. You can also cover the root area - as widely as possible, even beyond the drip line is best - with any organic (in the sense of natural, i.e.; shredded bark, wood, leaves, compost, etc., not in the meaning of chemical free, and not gravel, although in the right place, that's also attractive, and fulfills the other requirements) mulch, putting it no deeper than 4", and putting no mulch in the 2-4" next to the trunks of the plants. Mulch will eventually break down - one reason to use it, others being limiting weeds, keeping root areas cooler, and keeping moisture in the soil - so you will need to add an inch or so per year, if you do begin. Mulch also breaks the force of water, so is more likely to stay put, and not wash away like your soil did. If you are planting on a slope, avoid pine bark - it tends to float and wash away. You can use a "natural" fertilizer, such as cottonseed meal or alfalfa meal, compost or one of the Organic fertilizers (of whatever brand you chose; Espoma is a good one, 'though not all the types are purely Organic) lightly before spreading the mulch, as they are mild and not likely to burn. If you plant to use a chemical fertilizer, then I would spread an inch or two of whatever you use as a mulch down first, as the chemicals are much more water soluble, and might burn the exposed feeder roots. I trust this has answered your question, and not confused you further!

Mulching or planting within the dripline of any shallow rooted tree should be done with care, focusing on the needs of the TREE. Suffocation can result if roots are unnaturally "buried." Mulching should be done lightly and kept from contact with the bark. Any ground covers planted in the root zone should be drought tolerant. Keep in mind that any trees growing in lawn will be forced to shallow rooting because of the way they are watered.

The two photos are of two different trees. One is Bloodgood and the other is a Dissectum Red Select. So, the top photo looks okay and in the 2nd photo, the finer roots should be covered and mulched more? It just poured rain again! I haven't been out there yet, but I'm sure more soil has washed away. I took a chance and purchased these two little trees at Aldi for $12. I've been trying to baby them and, so far, so good. Thank you both.