Do electrons always have a probability of being somewhere? In the same way as when they surround a nucleus? How about when electrons go through wires or are ejected as beta particles? Do they still only have probabilities of being somewhere, or...?
 A: 
Do electrons always have a probability of being somewhere [in] the same way as when they surround a nucleus? 

Yes. Of course they don't have a probability a of being somewhere when surrounding a nucleus, they have a frequency of being found somewhere if measured, which is different. You can get a full probability too, but only if you specify even more measurements (a maximal commuting set of observables).
So the point is that you can't assume they have a position, or even have a position with a certain probability. Not unless you are going to measure it (and even more technically, you could assume they have a position but then you couldn't assume they have a momentum or a spin).
So they don't have a probability of being, they have a probability of being found when looked for. It might seem like a meaningless distinction, but it matters when you consider correlations between different measurements.

How about when electrons go through wires

Same deal.

are ejected as beta particles? 

Now it is trickier because they are being created. Again though, if you describe/do a measurement you can get the theoretical/experimental frequencies of various results. But sometimes the frequency will be to find no electron. And there is another complication during particle creation which is that really you only have frequencies of there being an electron somewhere you can't tell which electron is there. So even when the decay hasn't happened and that electron doesn't exist there is a nonzero frequency you find one of the other electrons in the universe.

Do they still only have probabilities of being somewhere, or...?

They don't acquire a well defined position in any situation, except maybe after a perfect position measurement, which would require access to infinitely large energy states.
A: 
Do electrons always have a probability of being somewhere?

It's lies to Pies again I'm afraid. We make electrons (and positrons) out of light waves in pair production. We can diffract electrons because of the wave nature of matter. And waves are not point particles. They are extended entities. They do not exist at one point only. So all that probability-of-finding-a-point-particle stuff is just lies to children I'm afraid. 

In the same way as when they surround a nucleus? 

In atomic orbitals electrons exist as standing waves. They don't exist as point-particles. 

How about when electrons go through wires or are ejected as beta particles? Do they still only have probabilities of being somewhere, or...? 

They still have a wave nature, hence electron refraction. And it's quantum field theory, not quantum point particle theory. Everything is fields and waves, wherein a wave is a field-variation and/or a field is a standing wave. The electron is not some point particle that has a field. The electron's field is what it is.   
