Our dirt is VERY dark and soft, it is not hard & compacted, is it too soft?

No.

Could the rain have been too much for the grass?

Probably not, but if the new grass was long and got matted down by the rain then that can be detrimental to it. You can also get some washout on slopes, and seeds can get moved around by lots of rain too.

We didn't properly "aerate" before seeding, would that make it not stay?

No, aerating isn't necessary.

The old tree leaves a lot of branches/leaves so I was trying to "lightly" rake them up so the ground would get the most sunlight etc. Was this wrong?

No, unless your raking was uprooting grass seedlings, but leaves left on new grass can lead to problems too. A leaf blower is usually the best way to handle leaves on new grass as it generally doesn't uproot seedlings.

With there still being some new grass still there & the older grass, would it be worth it to properly aerate & re seed "properly"?

The main issue to having grass there is that tree. Most shade tolerant grasses need a few hours of direct morning sun, and will do even better with 4-6. The grass also competes with the tree roots to some extent, often times you will see grass thin at the base of trees even if they are getting enough sun due to this competition, although some trees have less effect due to their root structure. The best solution is to remove the tree, stump, and major roots if you want nice grass there, otherwise it will probably always be a marginal stand. I have a somewhat special grass in my deep shade back yard that is very shade tolerant. It does well till about now, then it starts to thin till fall. Fortunately it is an aggressive spreader, so it comes back after the leaf drop in fall, and before the leaves come in the spring. If you were to seed again I would probably wait till late summer/early fall and use fine fescues.