Not that there's anything wrong with this: I've been watching DvD on Pluto TV, and it's occurred to me that at least a couple of episodes were adapted pretty freely by the writers at Frasier:

On DvD, "Big Max Calvada" (Sheldon Leonard) leans on the Alan Brady writers to create an act for his nephew, who is trying to break in as a comedian. In Frasier's "A Word to the Wiseguy," Jerome Belasco calls in a favor to make Frasier counsel his girlfriend. The key to the borrowing is that in both cases, the gangster actually wants the experiment to fail, so that his loved one will give up and get on with his/her life.

On DvD, "The Meershatz Pipe" Rob gets sick and freaks out that Buddy might take his job. On "Frasier Crane's Day Off," Frasier gets sick and freaks out that Gil might take his spot. The key to the borrowing is that in each episode, there's a scene where a relative weakling (Laura/Martin) wrestles the sick guy back into bed, forcing him to concede that he's really sick. Edit: Also, Mary Tyler Moore is one of the guest callers on the show.

Still editing: Carl Reiner was a guest caller in the early "Selling Out" episode of Frasier. His "problem" sounds a bit like Rob and Jerry arguing what to name their boat in the "Ballad of the Betty Lou" DvD episode.

Were there other episodes that seem like they were copied by Frasier, or other shows?

So plagiarism rears its ugly head once again in the entertainment industry: what a surprise in a domain of work where deadlines are demanded and mere mortals have run out of ideas.

By the way could you tell me who Alan Brady is and what the significance of his mane might be?

I never thought about it before, but there were a lot of similarities between Frasier and The Dick Van Dyke Show. Both shows focused on the lead character's home and work life and both had a show within a show. Both shows starred an actor who was a gifted physical comedian, could do witty repartee, and could also do a little song and dance routine when necessary.

Alan Brady is the star of The Alan Brady Show, the show-within-the-show of The Dick van Dyke Show. It is ambiguous as to whether "Alan Brady's Hair" refers to Alan's long-lamented natural locks, or to the collection of very natural-looking toupees that sit on Alan's desk.
and his name was Alan Sturdy in the pilot. They were going to keep that name but Mory Amsterdam said that "Alan Sturdy" sounds like "Alan's dirty" so Carl Riener decide to change "Sturdy" to "Brady"

That's interesting. I haven't watched Dick Van Dyke in ages so it's hard for me to compare it to Frasier but I think it's a compliment to the show if the writers did borrow a few things. Dick Van Dyke is a classic!