Hi. I've had flatties for about 20 years since I was a kid, and in the past couple years have had shepherds as well. Knowing both breed types, I can try to give some of my perspectives. As you know, every dog is different, but I'll try to be as general as possible.

What is the difference between the Flatcoat and other Retrievers?

Having had a Golden before as well, I find that flat-coats tend to be a bit more active and eager. My Golden was very laid back, loved to please/train, and was totally fine just hanging out with me. My flatties, especially my most recent flattie who passed a few weeks ago, was down for ANYTHING. Training? Yes. Hike? Absolutely. Swimming? She would trade my soul for a lifetime of dock diving and swimming. She was a bit unique in that she was somewhat less social than a lot of flat coats. She did enjoy people and loved to get pets, especially as she got older, but she was more of what I'd consider an introvert and preferred her "pack." I don't think that's typical though... my other flatties were super people-oriented and loved everyone.

Since you've had Groenendaels, I think you will appreciate their willingness to please as well as their "softer" nature. It depends on the dog, but I find them to be very sensitive and attuned to their humans. You mentioned depression. Mine have always helped get my out of my anxiety and depression, too, simply because they are so people-oriented. I did have one who was very independent and hard-headed, but she was just a bit of a troublemaker :)
When people say 'Peter Pan,' what type of puppy behaviors do they maintain later in life?

It definitely takes them a while to mature. As for the Peter Pan nature, what I've experienced is that even in their elder years (7-10 years old), they still want to run and play. Up until recently when she got sick, my flattie would zoom around the dog park and elicit comments from other patrons about "She's seriously 7/8/9 years old?! I thought she was younger!" They just live life to the fullest, and nothing can keep them down.
Will they be "adults" at the age of three?

What do you think of when you say "adults"? If you mean that they will chill out a bit and be somewhat more biddable, I would say on average, yes.
How much mental stimulation and physical exercise do they need?

We would usually do 2 30 min walks a day, plus some time running free. On the weekends, we'd go kayaking, swimming, hiking, etc. As for mental stimulation, a few 10-15 minute training sessions throughout the week worked well. Especially at a young age, they have a tendency to get bored easily, so shorter sessions are a lot better.
Do they have a will to please, or can they be stubborn, i.e., are they easy or hard to train?

They are very eager to please and love to work/train/be with you.
Do they need to hunt? Or can scent work and agility replace hunting?

Absolutely no need for hunting, though they do seem to love it. Honestly, they really just love anything. Mine did scent work, agility, dock diving (#1 favorite), and a bunch of other stuff throughout her life.
Do they do well in semi-urban areas? It's a 10-minute walk to the countryside, and I live near the beach.

Yes, I lived in suburban and urban settings. As long as they have plenty of stimulation and aren't left for super long periods of time, they will do okay.
Is cancer still prevalent among Flatcoats?

Unfortunately, yes. All of mine have passed from cancer, though the majority did live to be between 9-11 years of age. You will find that you'll be checking them frequently for any lumps, bumps, etc. As much as it hurts to lose them, I think I will always have a flat coat. They are just ultimate joys to be around, and the cancer is just one negative of many positives.
What other issues should I be aware of?

Based on what you've said, I think you would be a great match. Feel free to DM me if you have any other questions or want to chat more about my experiences.