If firearms were to exist in Tamriel, they would be heavily regulated (the Legion would most likely be the only place they would be found).
I think you over-estimate the benevolence of Legion soldiers. If they became standard weapons, at least for a new division of the Legion, then you can bet they'll be on the market in a short time.
You could make several more reliable weapons (i.e. swords and the like) for the cost of one firearm, and thus reliably arm more individuals.
I respectfully disagree. The cost a a well-balanced, sturdy blade of even iron (and remember, the Legion seems to favor silver and steel weapons) will far exceed a simple wooden musket with a few small metal parts, and for the time it takes to make said well-balanced sword of metals, you can turn out mass-quantities of these muskets. That's why they became so popular, because a single bow takes a long time to make, whereas a musket can be easily made in the simplest assembly line and churned out in mass.
Why go to the expense of creating a "gun corps" when you would lose as much as half of them to misfires every time they went into combat (or even during training/practice)? You could arm and field twice as many troops (or more) with "conventional" weapons, and you wouldn't have to worry about those weapons exploding in the troops' faces.
I think you are over-estimating the amount of mis-fires these things would have in semi-trained hands, but I could be wrong.
Financially, it's just not feasible for the Empire to invest in guns (and it would have to be an Imperial investment). If the Legion really, really needed highly mobile weapons with a greater range than bows, the Emperor could always order the Mages Guild to supply staves (and filled soul gems to "reload" them). A magic staff would be far more reliable than a crude firearm.
As I said, muskets are more financially sound then swords and staves. Remember how expensive it was to recharge those damn things in OB?
Imagine doing that for 10,000 Legionnaires after every battle (if they became a standard weapons). The only expensive thing about a musket is probably the research they would have to put into it, but once the design is there, they become incredibly cheap to produce in mass, unlike staves and
good blades.
Plus, if they had to rely on the Mages Guild, a
non-Imperial organization to do this, that would become way more pricey then government-run factories for muskets in which they have full control over prices, whereas the Mages Guild would undoubtedly monopolize the recharging of Legion staves and drain the coffers dry.