Lots of questions

Your replies.
Depending on models and what one you get. Some have maps, others do not.
Mountain Biking:

• If I go on a mountain bike ride, can it record the route so that I can then come back to the same place a second time and have the Garmin guide me through the same ride? YES
  
• If so how far off the original path can I go before it starts telling me I'm going the wrong way? A foot? Ten feet? 200 feet? A mile? Within a few meters
  
• Will it tell me the total distance of the ride, average speed and elevation change? Yes
  
• Is there way to attach it to my handlebars so that I can see where I am the whole time? Yes
  
• Can it stay turned on so the display is visible for an entire three hour ride? Yes, up to 11hrs
  
• Is there a way to have the route that I rode displayed on a topo map? Not on here, maybe 3rd party or google earth
  

Road Bike Riding:

• If I go for a road bike ride, can it record the route and then translate that into turn by turn instructions for the same route the next time I ride it? Yes
  
• Can I then upload that route to the internet so other GPS users can then download it to their Garmins and use it to follow the same route? Yes
  
• By the same token, let's say I'm going to visit some family in Europe and I want to bring my bike, is there a way I can go online, find good road riding routes that people have already uploaded, download them to my Garmin and then take it on a ride and have it give me turn by turn directions so that I can concentrate on riding and not trying to figure out where I am the whole time? Yes
  

Skiing:

• Can it record the total number of vertical feet that I skied in a day? Yes
  
• Can it tell me the current elevation? Yes
  
• Can it be exposed to extreme cold for an entire day? Yes, but battery life suffers when cold. I have it in a MP3 insulated case when its below 0C
  

Driving:

• If I'm driving somewhere can I punch in the destination address and then have it give me turn by turn driving instructions to the destination the same way a Garmin Nuvi would? NO
  
• Can it display points of interest? NO
  
• Tell me where how far to the closest rest stop? NO
  
• Is there a way to mount it on the dashboard and charge it with the cigarette lighter? Not really but you could jerry rig it
  
• Can it store common destinations so that I can easily use them again later? Yes
  
• If there's a traffic jam can it suggest an alternate route? NO
  
• Can it tell me ETA? NO
  
• Can it do all this in North America and Europe too? Depends on model and maps loaded
  

Finally, I've seen various iPhone apps that can do a lot of this. Why not just get an iPhone instead? What can a Garmin do that an iPhone can't?
HR, Cadence, designed for sport.

The unit is not like a car based GPS, get your hands on one or walk though some info on here and user manuals for ideas as to what will work .

KIN

Mountain Biking:
If I go on a mountain bike ride, can it record the route so that I can then come back to the same place a second time and have the Garmin guide me through the same ride? yes
If so how far off the original path can I go before it starts telling me I'm going the wrong way? A foot? Ten feet? 200 feet? A mile? it depends on your GPS reception, but generally not far
Will it tell me the total distance of the ride, average speed and elevation change? yes
Is there way to attach it to my handlebars so that I can see where I am the whole time? handlebars for most, stem mount for the Edge, and wrist for the Forerunners
Can it stay turned on so the display is visible for an entire three hour ride? yes
Is there a way to have the route that I rode displayed on a topo map? if you get a mapping receiver, yes

Road Bike Riding:
If I go for a road bike ride, can it record the route and then translate that into turn by turn instructions for the same route the next time I ride it? the GPS itself won't do that, but various software and websites can do that
Can I then upload that route to the internet so other GPS users can then download it to their Garmins and use it to follow the same route? yes
By the same token, let's say I'm going to visit some family in Europe and I want to bring my bike, is there a way I can go online, find good road riding routes that people have already uploaded, download them to my Garmin and then take it on a ride and have it give me turn by turn directions so that I can concentrate on riding and not trying to figure out where I am the whole time? sort of. to get those turn-by-turn directions, you will need to do some work in the computer first

Skiing:
Can it record the total number of vertical feet that I skied in a day? if you get one with a barometric altimeter, it will, but realize that the inherent error with a barometric altimeter is usually a bit more than the GPS itself. many programs don't care if you have a barometric altimeter and will overlay your track onto a digital elevation model (DEM) and calculate your elevation change that way. but to get anything in the GPS itself, you need a barometric altimeter
Can it tell me the current elevation? if you have a barometric altimeter, yes, as long as the sensor is properly calibrated. if you do not have that sensor, then you need a 3D position fix
Can it be exposed to extreme cold for an entire day? it will depend on the batteries you have and how you keep the GPS. alkalines STINK in the cold, but lithiums are better. Also, if you keep the GPS inside your coat, it will not get so cold, so batteries will not be affected as much.

Driving:
If I'm driving somewhere can I punch in the destination address and then have it give me turn by turn driving instructions to the destination the same way a Garmin Nuvi would? this requires the City Navigator software, but many models will do this
Can it display points of interest? yes, but that depends on what software you have installed
Tell me where how far to the closest rest stop? not really/yes. in order to have it do that, you need to search for rest stops, and it will give you a list of rest stops and their distances/directions. BUT, doing this on a touch-screen nuvi is bad form if you are driving at the time, since it's quite distracting. doing this on a model with a trickier interface will be impossible if you are driving. either have the navigator do it or pull over.
Is there a way to mount it on the dashboard and charge it with the cigarette lighter? you can get dash mounts and 12V power adapters for many models, BUT, that 12V power adapter will not charge the receiver unless it has a rechargeable battery like an Edge. all of the handhelds take AA batteries and cannot be recharged in the GPS, even if you use NiMH cells.
Can it store common destinations so that I can easily use them again later? you can save a waypoint anywhere for later use in most models
If there's a traffic jam can it suggest an alternate route? only higher end Nuvis can do this, and they require an XM Traffic subscription to show traffic information. most models will not even give you an option to seek an alternate route for other situations, either (bridge out, power lines down, etc)
Can it tell me ETA? yes
Can it do all this in North America and Europe too? you have to buy the software. Europe software is often spottier than US software from what I hear

It sounds like you want a GPS for a lot of things. A GPS that can do all those things will not be good at any of them. Ideally, you'd get a function-specific receiver for each intended use. At any rate, at least a car GPS and an outdoor handheld.

Thanks for your answers. Both were really helpful.

-chris