Can interconnect cables improve sound stage height, width, and depth?

Absolutely. You need a bigger house? Well good interconnect cables can alter space time so your house can be as big as Noel Lee (after all the money he made scamming everyone)?

The correct answer is that it all depends.
The effect of cabling choice is system-dependant.
At the risk of poking the bear, I have a theory that the naysayers who claim "wire is wire" or "assuming capacitance and resistance are within a certain range-wire is wire", have systems that don’t reveal changes in cabling. It makes perfect sense that the folks who don’t want to spend the money on more expensive cabling or listen with an open mind are the same folks who buy lower level amps and preamps and speakers thinking that all amps sound the same and that preamps are nothing but input selectors with volume controls and that inexpensive loudspeakers are plenty good enough.
At even further risk of poking the bear(s), damn there are a lot of dumbasses on this Board! It blows the mind! Exhibit A is the chicken crying that the sky is falling because one of Kalman Rubinson’s reviews did not include measurements from which said chicken falsely concluded and declared that Stereophile has now globally dispensed with measurements. This is stupefying stupidity. In that same thread, a very cynical but unfortunately likely correct person concluded that 97% of us humans just don’t want to use our brains any more than we absolutely have to.
Can anyone imagine how difficult it must be for a bricks and mortar salesperson to deal with the public when this is the status quo? How in the world can a salesperson (and let’s face it, 99.999% of them are male, so "salesman") possibly explain to a price-conscious customer that for the most part, you do get what you pay for in audio? So for this reason and many others, there are fewer and fewer bricks and mortar stores and now we have even less educated lemmings who buy over the web and express their views based upon what little they think they know.
So to the OP, I advise you, and you can take this as worthy of thought or not, that the answer is unequivocally "yes" that better cabling can improve the perceived soundstage width and depth, but it depends on so many other factors such as the equipment design and quality and whether the room is properly set up and whether the source is good among others. Without going into specifics, I recently completed a six month long experiment of various IC’s and SC’s and heard tremendous differences in the soundstage width and depth among other things with each change of cabling. But my sound system is not yours.

A change in interconnect cable certainly can improve imaging, but, if you are having problems in that area, it is not likely to be a cure.  In other words, you really have to get everything right to get terrific imaging and there is no way that making changes in one part of your system can compensate for inadequacies in others. 

First and foremost, you have to get speaker placement and placement of your listening chair correct.  Taking care of this can turn almost any speaker into an imaging champ.  I've heard remarkable imaging from a vast variety of speaker types, sizes and shapes, and it mostly comes down to correct placement.  This is too broad a subject to get into the details, so you need to do some research, and mostly, a lot of experimentation with moving things around.  Room treatments can help, but, placement is the key.  Also, it is simply not realistic to expect great imaging in anything but a tiny listening window unless you are using omni-directional speakers.  Even then, the ideal spot will only fit one listener (the two dedicated listening rooms that had a wider sweet spot than one person were gigantic--around 30' x 55').

Electronics certainly matter, and the best gear at giving one a sense of a sound-field that is completely enveloping the listener tends to be tube gear, particularly low-powered tube gear (e.g., single-ended triode amps).  Whether or not this is an artificial thing with tube gear, it certainly is seductive.

Once you get everything right, it is pretty easy to hear differences in cables and some wire tend to do certain aspects of imaging right.  But, as with anything in audio, any choice tends to involve some tradeoff (no one thing is superior in all aspects of performance).  For example, some wire that sounds very clean and detailed, will tend to be very precise in terms of image placement, but, that type of wire tends to be tonally thin and this detracts from the sense of the sound filling the entire listening space.  Finding the balance that fits your priorities is the key.