A function generators is the jack of all trades of signal sources. It can do sines, square waves and triangular waves (and often variants like ramps and pulses) from audio frequencies up to 1MHz or so. They're not the best at any single aspect, but they are the most versatile. If you only need pulses, pulse generators offer more features and bandwidth. If you want very pure sine waves, an audio oscillator will have lower distortion. If you want a really stable frequency for RF work (eg. as CW), a signal generator has more bandwidth and a more stable output. But if you don't have any special requirements, a function generator will do (some of) all that for less money.
Sines are useful for sweeping, bandwidth and gain tests. Square waves let you see both the low and 'high' frequency response of a circuit at the same time. Triangle waves are useful for testing cross-over distortion and clipping.
5: If you don't need a function generator, great, you just saved yourself some money
. I've used them as clock source, testing counter IC's, testing amplifier bandwidth, testing dynamic range (clipping), measuring capacitance/inductance, injecting signals in circuits and much more.
1,2:It does depend on your intended use. There are basic function generators that barely exceed the audio frequencies, and really expensive function generators that do arbitrary waves and sines up to 80MHz or so (Agilent 33250A). Both have their uses. As basic, I would probably look at something that does at least sines up to 1-5MHz or so, and also square and triangular waves (these often have a lower max frequency, read the specs carefully). Maximum voltage is usually something like 20V in open circuit, I more often bump against the lower limit than the upper limit, so I value something like a built-in attenuator. The output impedance is usually either 600 ohm (audio standard) or 50 ohm (rest of the world). If you're not into audio, 50 ohm is most useful, although this probably doesn't matter much at these frequencies.
3: I don't consider a built-in counter important, but it depends on your other equipment. If you have another counter or digital scope, you can use that. Plus the frequency dial should have a pretty good resolution, and a function generator isn't designed as stable, accurate frequency source. If it does have a counter and you don't have anything else to measure frequency, it might be nice if you can feed an external signal into the counter. Not sure what you mean with the kind of isolation? Some function generators float up to 50V or so, this can help against ground loops, but I don't consider this very important for my work. Not sure what digital vs. analog means I haven't seen many analog counters. There are digital function generators, these use DDS (direct digital synthesis). This creates more stable signals with less distortion, but is more expensive and more recent (you're less likely to find a used DDS function generator).
4: Depends on how you define better
. I haven't seen many good portable function generators, but I haven't searched either. Most are bench style, so unless you have a specific need for portable, you should probably get a bench one. I'm not a fan of portable instruments for lab use. With bench instruments, I don't have to change batteries, I can just stack them, they often have better controls and displays, and they just stay in their place (unlike the DMM stand issue that Dave just published).
Dave did a review of a (GW-Instek?) function generator in one of his early blogs, I think his opinion was somewhat mixed (very poor phase noise?), but there's probably some useful info in there even if you plan to buy something else.
You might consider a used function generator from something like ebay, not sure how it would compare to price/performance to current items. Wavetek was a well-known brand for function generators.