I haven't read all the responses but since my husband and I have done three estates, I'll put in my two cents. (He was the trustee/executor but I know how to do the paperwork!).
I'm not sure how Florida works but in California, all three people had trusts. With a trust, you get copies of the death certificate, go to a lawyer who creates a Certification of Trust. Depending on how the trust is set up, a new trust is created (kind of like a survivor trust) and you are the trustee of that trust. The new trust gets a tax ID number and all the assets can be moved into the new trust. You will have to send the death certificate and the Certification of Trust to the firms that hold the accounts who will then let you move the assets into the new trust (or sell them, etc.).
The house would remain in the old trust. It would be sold pretty much just like any other sale (with you as the successor trustee) and then the proceeds would go into the new trust.
With a trust, it is really quite simple. I am on couple forums with you and you seem financially savvy so I think you will be able to handle it pretty easily.
Of course, it was years ago that we did this so things could have changed. If he doesn't have a trust, it might be more difficult and things in Florida could be totally different.
In California, if you don't have a trust, selling a house has to go through probate and the sale has to be approved by a judge. That can take a while.
Re the stock certificates, does he have a brokerage account where he could deposit those now before he dies? That seems easiest to do.
Good luck and I'm sorry you are going to lose someone.