The profile is not whitebalance, and generally speaking should not affect white balance. Also, cloudy vs sunny should not vary much, as that's white balance.
The profile is more about fixing color errors in individual colors due to either the spectrum of the light, or the sensor. As an example, if your lighting source (let's say mercury vapor lights) makes the blues over-saturated, but reds dull, building a profile under that light will tend to de-saturate the blues a bit, and add saturation to the red (it's probably not "saturation" but a change in hue, I'm not sure, I am speaking more how it appears than what it does). It won't make it perfect, as some light is just too bad, but it makes it better. I can see big differences in certain colors (greens, blues especially) on my Nikon shots in arena lighting, for example.
If you build a daylight profile, and alternate between applying it and using the LR default, you can expect to see only very subtle changes, because the Adobe default profiles are pretty good, and your sensor is likely pretty good. But there should be some minor difference.
On the other hand, build a profile for more unusual lighting, and try again. You may see more distinct difference.
The other place you can see an impact is if you have two different cameras, either different bodies, or different vendors. There's often a difference in color "look" between them, and the profile will tend to synchronize them. Note however, that before comparing that effect, you need to match their white balance. I've found different bodies have significantly different white balance even from the same vendor. Profiles are not white balance (keep thinking that).
As to your question about what to do with white balance when building the profile, since the image used for the profile is raw, it should not matter, though I have not tested to see if the profile software uses the AWB settings in the raw file. That's an interesting question, I have always used AWB, not a specific white balance.
When I build profiles it is usually for specific lighting, e.g. at a baseball stadium, a soccer field, etc. What I'll do is shoot the passport in daylight, then shoot it under the lights, and build a dual profile. That way as it goes from daylight to dark (lights) it still works properly, interpolating (or something). It's quite important to get the passport where the subject will be. Don't (for example) set it on nice green grass and take a shot, or anywhere there's a strong reflections from any colored object (unless that object is going to be the same for all your shots). Also, I've found it pretty important to under-expose a bit so no channel is saturated, otherwise when you build the profile, it will err out. Best is just take a bracket of say 5 stops or so, then you know you have something to use.
I also have a "normal" that I built with daylight and with incandescent for other places, though the whole idea that incandescent is "inside light" is largely gone, with the advent of LED's. LED's vary a lot, and if you are using them (say in a studio) I would make a profile from them as well.
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