How often would you switch to HDG instead of NAV? Multiple times per flight? A couple of times a month, never unless the situation called for it?
Well, "never unless the situation calls for it", happens to equate to multiple times per flight.
There will normally be a SID loaded, and the aircraft will transition to NAV shortly after take off. But, in many places, you won't complete the SID, with ATC giving vectors for a while before clearing you back onto the planned route. On the arrivals side, they vary from almost complete disregard of the STAR, to following it exactly (Melbourne is probably best at this). You simply change as needed.
If you do switch to HDG, obviously that turns off LNAV, but what about VNAV is that kept on in HDG mode?
LNAV and VNAV are Boeing terms. In a Boeing if you select HDG, VNAV will remain engaged, but in an Airbus, as soon as you select HDG, you also lose the vertical mode (in a climb it goes to 'open climb', and in a descent to 'v/s'). There are arguments both for and against both systems.
If VNAV is also turned off, how is vertical navigation done? Alt Hold or V\S (as infrequently as it's used)? Manually?
Well, if you can multiply by 3 or 4 then you can work out the altitudes/distance in your head (and you have to keep an eye on it anyway, as the computers quite regularly lose the plot with regard to vertical nav). In both types, deviation from the planned vertical path is displayed one one of the performance pages of the FMC. All of us have a few energy gates that we need to hit too...250 knots, 20 miles, and 5000 feet...works for everything.
Altitude hold isn't used often. It's a mode that stops the climb or descent at the point you are now. It's a bit abrupt from the passenger perspective as it may be associated with a large power change. Actually the most common use is in the sim, at the clean up stage after an engine failure. V/S on other hand, is used a lot on descents, very rarely on climbs (as it has drawback that can be dangerous). Open descent is used if you want idle power. It would be a very rare flight in which all of the modes....open, V/S, and managed...aren't used, with multiple changes between the modes.