Why don't (or why do) current carrying wires attract a stationary charge placed at a distance? I've learned that moving charges produce magnetic fields which in turn affect other charges in motion. After seeing explanations that point to special relativity, I am kind of confused. Can ALL magnetic fields be accounted as some kind of electric field from a particular reference frame? 
And if there is relative motion between the electrons of the wire and the charge at rest(from the lab frame), then will it not experience a magnetic force from the electron's reference frame? I am not sure if that is the actual case, so even if the stationary charge is attracted to the wire, can it be accounted as an electrostatic force from the lab frame due to length contraction and as a magnetic force from the electrons POI?
I am not even completely clear with even how to phrase the ambiguity I have in my mind.  Detailed answers are very much appreciated :)
 A: 
Can ALL magnetic fields be accounted as some kind of electric field from a particular reference frame?

No. Relativity really tells us that electric and magnetic fields are on an equal footing. In some situations, you can find a frame where there's only an electric field. In others, you can find a frame where there's only a magnetic field. But most of the time, you can't do either.

And if there is relative motion between the electrons of the wire and the charge at rest(from the lab frame), then will it not experience a magnetic force from the electron's reference frame? I am not sure if that is the actual case, so even if the stationary charge is attracted to the wire, can it be accounted as an electrostatic force from the lab frame due to length contraction and as a magnetic force from the electrons POI?

I'm not sure if this is getting at your confusion, but recall some basic examples in relativity. For example, suppose that in your frame a spaceship passes by you. In your frame, this can happen really quickly because the spaceship is length contracted. In the spaceship's frame, it happens really quickly according to you because your time is dilated. So what's really going on? Is it really time dilation or is it really length contraction? Of course, the point is that the two frames are on an equal footing. Time dilation in one frame can be equivalently described as length contraction in another, and neither is inherently more correct.
Similarly, in some situations, what can be described as a magnetic force due to motion in a magnetic field in one frame, could be described as an electric force due to an electric field in another frame. In each individual frame, absolutely everything works as usual: Maxwell's equations are true, the Lorentz force expression holds, and so on. So, for example, in a frame where a charge is still, it experiences no magnetic force, even if it might in a different frame where it is moving. The description of what is going on changes between different frames, but neither frame is more "correct". 
Saying that magnetic forces are "always really just because of electric forces due to a charge imbalance due to length contraction in a different frame" doesn't make sense. It doesn't work in general, and it's kind of like saying "time dilation doesn't really exist, only length contraction does". It's actually the exact opposite of the spirit of relativity.
A: Forget special relativity.
In the lab frame the wire is charge neutral so has no electric field.
However, in the lab frame there is a current. This current produces a magnetic field.
That means there will be a magnetic field at the location of your test charge. However, you specific a stationary test charge. Recall that the magnetic Lorentz force on a charge is proportional to velocity. So even though there is a magnetic field at your test charge, it feels no force.
That is to say, a charge is NOT attracted to or repelled from a current carrying wire.
edit:
Now let's change the reference frame into frame moving parallel to the current carrying wire (frame might be moving with or against the majority charge carriers). Suppose the current is generated by positive charges moving in the $+z$ direction within a lattice of negative charges. Speaking qualitatively a few effects will happen.
(1) In the new reference frame the positive and negative charges in the wire experience some amount of the length contraction. The length contraction will differ between the two types of charges because the + and - charges have a relative velocity. This difference in length contraction leads to a resulting difference in the charge density for + and - charges. This difference in charge density means the wire appears to have a net-local charge in this moving reference frame. I believe This net charge could be either positive or negative depending on which way we've boosted resulting in E-field lines pointing towards or away from the wire. This E-field puts a force on the test charge.
(2) In most boosted frames there will still be a net-current through the wire. This current generates a mangetic field at the location of the test charge. However, in this moving frame, the test charge now has a velocity. The test charge is now moving with some velocity through the magnetic field. If you work out the right hand rule you will see that the Lorentz force on this test charge is either towards or away from the wire.
My claim is that the electric force from the length contracted charge densities and the magnetic force exactly cancel out so that there is no force towards or away from the wire no matter how fast you boost along the current carrying direction. I don't prove it here, but someone with more time could more carefully go through all the math, boosts, negative signs, etc. and find the expected result.
A: Let me consider an extreme case: Does a current carrying wire attract a stationary charge placed at a distance if the current in the wire is zero? In that case there is no motion, relativity theory is irrelevant, so what happens?
The electric field of the stationary charge polarizes the charges in the wire, so it looks like the stationary charge will be attracted to the wire.
