I've read that ICAO defines Class F airspace but the FAA has chosen not to use the airspace class in the US. What is the ICAO definition of Class F airspace and how does it differ from other airspace classes? What countries use Class F airspace? Why does the FAA only use A-E and G?
ICAO Class F airspace is a bit of an odd duck (and the US FAA is apparently not the only agency that thinks so - from a quick check on Wikipedia it seems more jurisdictions ignore class F than implement it. They only mention Class F as being in use in Germany and the UK).
From a functional/regulatory standpoint Class F is a sort of hybrid between "Class E" controlled airspace and "Class G" (fully) uncontrolled airspace:
Class F airspace is designated as "uncontrolled", but you can get ATC clearances in Class F airspace just like in controlled airspace. The catch is they're "advisory only" (so you don't have to comply with them, and all the weight is on the pilot's shoulders ). Similarly ATC will provide separation services to IFR flights in Class F airspace, but they do so "where possible", which means they might tell you they can't provide that service and you're on your own with see-and-avoid.
The general use case for Class F airspace seems to be allowing IFR flights to operate in "uncontrolled airspace".
As a US pilot (and thus having no experience with Class F Airspace), it sounds like the services provided in Class F airspace are effectively the equivalent of VFR Traffic Advisories ("Flight Following"): Controllers can give you advisories and make suggestions, but all responsibility is ultimately still on the pilot's shoulders, and you can politely tell the controller to get stuffed if you don't like their suggestions (without needing to declare an emergency to vest yourself with that authority).
In the US we already have the Flight Following system and culture set up for VFR folks (and generally speaking it's available nationwide, irrespective of airspace class, as long as the controller can get you on radar and isn't overworked). We also have a pretty robust Class E airspace nationwide.
Because of this the FAA probably sees no need to designate specific "Class F" regions where IFR flights can get the equivalent of "Flight Following": If you're VFR you can usually get flight following if you want it, and if you're IFR you should have no trouble staying in controlled airspace (Class A, B, C, D, or E) for your entire route of flight.
Class F airspace is often used in the UK as a kind of "GA airway." It designates preferred paths with a advisory ATC service that GA traffic can use.
For example, there is a class F route defined between the north west of England and the Isle of Man. Having a preferred route and an ATC service makes sense for that route as its an expanse of open water. It simplifies search and rescue operations.
As voretaq7 said, it's a bit of an odd duck, and I don't know how it works in the UK but in Germany F is only to allow IFR approaches/departures for uncontrolled airfields. So basically, as soon as an airfield in G has an IFR approach, they get a F airspace around them, which in turn is usually only active when there's an IFR approach in progress. As soon as it's completed, it reverts to G (depicted on the sectional as "F(HX)", meaning it's not always active). So in order to determine if F or G rules apply, you need to ask FIS or AFIS (or approach, but then you're probably IFR already, so you don't need to care).
VFR "flight following" is not related to the airspace and is provided by FIS in E, F and G airspace, as long as radar coverage is available pretty much. In D and C it's called CVFR and is provided by approach (or "Radar" as they like to call it). It works the same way when shooting an approach into a G airfield in the US, you're only cleared for the approach, as soon as you enter G, it's up to you to see and avoid, as far as I know.
The point of having an F airspace is that the visibility requirements are different for VFR aircraft, where you have to stay a mile away/thousand feet from clouds, whereas in G they only need to stay clear of clouds. Allowing IFR approaches in G has the potential of allowing an IFR aircraft pop out of a cloud and not have even a mile or thousand feet separation from the next VFR aircraft. Note that these extra VFR requirements are specific to Germany and deviates from the ICAO definition of airspace F.
Germany is in the process of discontinuing the use of airspace F for IFR approaches into uncontrolled fields, in accordance with EU regulation 923/2012 "Standardised [sic] European Rules of the Air", or SERA. Once SERA is in effect, IFR approaches will be allowed within airspace G in Germany as well. Current airspace F will be replaced with airspace G and a so called "Radio Mandatory Zone", or RMZ.
In Canada anyway according to the CARS (Canadian Aviation Regulations) Class F airspace is "airspace of defined dimensions within which activities must be confined because of their nature, and within which limitations may be imposed upon aircraft operations that are not a part of those activities." Basically it's airspace dedicated to a certain activity and you better make damn sure you're not going to be getting in the way of that activity before you enter.
Canadian Class F is better compared to U.S. Special Use Airspace, if you read the regs for it, it covers the Canadian equivalent of Warning, Alert, Restricted, Prohibited, Military Operation, and Controlled Firing Areas. Interesting that in the U.S. we do not depict,or inform about controlled firing areas, Spotter aircraft, and controllers, prevent non-participant aircraft from being harmed. But they are depicted on Canadian charts, and basically treated like prohibited, or restricted areas as needed.