When looking at finger prints they look for ridges, loops, arches and other patterns. Those ridges and loops are not found on other parts of the body (except for toes). They will look for toe prints or may find them, but how often would someone commit a crime barefoot? Generally if someone leaves toe prints around they will leave finger prints.
As for not using a print from another part of the body, they do do that. Forensics can use hairs, saliva, sweat, blood, etc and get DNA from it. That's a form of a print, in my opinion.
There aren't as many usable prints as you might think on objects such as public pay phones and rental cars. But the techs would take any print they could find that wasn't completely useless, run them all throw the database to look for matches. The prints probably wouldn't be usable in prosecution without corroborating evidence anyway, so all they could hope to get is a lead.
Here's a little reading for you about the subject:
http://www.essortment.com/all/fingerprintssec_rwkx.htmAs for the police dogs, they pick up the scent of humans. If you have a piece of clothing from the suspect, the dog would know the smell of the dead skin cells from the person that used it and that's what they would track. A little info about that is at the bottom of this page
http://hubpages.com/hub/K-9-Police-Dogs under "How Police Dogs do their work".
A K-9 unit is generally one of the first units on site, so there wouldn't be as many people around as you would generally assume. The suspect would have had to come from somewhere or be last seen somewhere, so that would give the officer a place to start with the dog. They go from there. The dog doesn't lead the handler to the suspect in 15 seconds like they always show on TV, it can take some time. They also usually have a specific area they know the suspect is in so it can be easier for the dog to seek out the suspect. Also, generally, the suspect will be sweating, breathing hard and scared. All things that the dog can use to help pick up a stronger scent or hear him or her.