Since Prince Philip has passed away, will Prince Edward inherit his father's dukedom?

As The Queen did not issue Letters Patent to alter the mechanism for succession to the dukedom, Charles has already inherited the title.  

In theory however, there is nothing in law to prevent her from revoking the title and recreating it for Edward after the official mourning period ends.  

No. The title passes to Prince Philip's heir, who is Prince Charles as per the Letters Patent of 1947 issued by King George VI.  Most of Charles's titles will become extinct when he becomes King including the Dukedom of Edinburgh. It is is expected that it will be re-created for the Earl of Wessex, which means it will not be in his mother's lifetime.

No, Charles has inherited it, as the eldest son.  However, when he becomes king, he is expected to recreate the title for Prince Edward.

When one Royal dies,  there is a great shake up of Royal titles, with some LOSING the titles as they advance  and others finally getting a slice of the pie- 

Surely  everyone will do well,  from the passing of Philip,  both monetarily and in the form of  "Moving on Up"  , as George Jefferson would say- 

Prince Charles is the Duke of Edinburgh. The thinking is that when he becomes king, the title will merge with the crown and cease to exist. Then it can be recreated for the Earl of Wessex, who would henceforth be known as the Duke of Earl. 

He already has a dorkdom!

Get a life......

Edward will be bestowed Duke of Edinburgh, which is different than inheriting the title. The eldest son of a duke usually inherits the title. Charles would inherit the title, but it was decided that since the future king doesn't need another title, that his youngest brother would one day assume it. Charles will let the title revert to the Crown so that Edward may be bestowed the title. Elizabeth may bestow her youngest child this title, or he may have to wait until Charles is king to be bestowed the title. The timing of the bestowal of the title depends on if special Letters Patent are issued so Edward could be bestowed Duke of Edinburgh during his mother's lifetime.

It all depends what's in the will. It is a bit early to be squabbling over the inheritance so they aren't talking about what's in the will yet and waiting for the lawyers to read it after the funeral.