Just wondering if ATC can do this or not. I'd also appreciate input on whether it's useful or a good idea, and how it looks in the real world.
They can. I often pick it up here in the Northeast since the airspace is fairly busy. Generally speaking I was taught that flight following is always a good idea. It is a useful tool to you in the cockpit and can provide advisories of fast moving hard to see aircraft as well as those that may be approaching to overtake you. Generally it helps to improve your radio skills and may reduce your work load ultimately.
As for who to contact, that can change. I had this very question during my training (asked here) and you can find the answer here. If you are departing from a controlled field you can sometimes ask the tower to set it up which they may or may not do. The Airport Facility Directory will have an "approach" and/or "departure" frequency listed for most fields which is generally who you call. If you are in the wrong area they generally send you over to the right controller.
Yes, you can request VFR flight following in Class E airspace and 90% of the time you will be flying in Class E airspace. There’s nothing unusual or difficult about it; you simply request flight following and ATC will usually approve it unless traffic congestion, emergencies, etc consume all available ATC resources for this.
Yes, as far as I know the entire US Airspace is covered by one center or another. It is the centers which provide flight following. For example on a recent flight from CYKZ ( Toronto Buttonville ) to KFRG (Farmingdale Republic on Long Island , NY), I asked CYKZ controller for flight following ( on the ground ) and she handed me off to Toronto Center. I again requested flight following and the request was granted. Toronto Center handed me off to Buffalo Center, then to Rochester center, Binghamton Center, Wilkes Barre Center, Boston Center, New York Center and finally to KFRG Tower. I believe if I didn't pick up flight following with Toronto, then I could have picked it up with any of the other centers mentioned above. In the JFK area I have picked up flight following with New York approach ( there are different frequencies depending upon whether you are going East or not). On a recent approach NY approach frequency was very busy and I ended up picking up flight following from Albany approach on my way to Montreal. These rules apply in US only. Canadian rules are different and I am still trying to figure them out.