I've come to understand that you guys have squadrons that are like a battalion level unit, groups are like regiments, and wings are like divisions. What I haven't been able to figure out is what the internal structure of the squadron is like. Are there smaller units that would be like companies?
Also, how many aircraft are in a typical squadron? And does this vary between different aircraft types (i.e. a 46 squadron vs. a Hornet squadron)?
Kinda...'cept, different. You're right that the hierarchies are roughly similar, but there are a lot of differences in what each level does. Plus, Marine Air Wings are organized and run very differently from Carrier Air Wings (CVW). In addition, there are Type Wings. Simply put, a CVW is an operational unit - the Wing trains, deploys and fights as one unit. Type Wings are administrative in nature; as the name suggests, they are responsible for all of a particular type of aircraft, insofar as maintainence, training and development.
A squadron simultaneously belongs to both wings. The CO (O-5) of a squadron reports to the Air Wing Commander ("CAG") operationally and the Type Wing Commodore on administrative matters. That's the carrier-based Navy; the shore-based Patrol and VQ guys have a Wing Commodore but usually only deploy as squadron detachments.
Navy Air doesn't use the term "Group". I'll let the Marines speak to how they use it.
Navy and Marine squadrons are organized pretty similarly: the squadron consists of departments (Operations, Safety, Training, Maintainence, Admin), headed by an O-4. The departments are broken down into divisions (run by an O-3), though usually only Maint is big enough to bother doing that. The Maint divisions are responsible for different parts of the airplane: Avionics, AME/PR (survival equipment and ejection seats), Airframes, Powerplants (engines), Weapons, etc. Bigger divisions will be broken up into shops or branches (run by an O-2).
So you can see, it's pretty much all organized for maintaining the planes and making sure the aircrew are ready to fly.
The number of planes in a squadron varies widely depending on type and mission. E-2 and Prowler squadrons only have four airplanes, while a Hornet squadron will have a dozen or more. Helo squadrons can have anywhere from six to a couple dozen airplanes.