Would poorly strung strings ever hurt your guitar?

That one-wrap-around-string will more than likely not stay in tune because it will slip around the tuning peg . All is not lost , single strings are sold separately . I would suggest that if you have trouble keeping that string tuned , buy a replacement and be more careful next time .
As far as the string action being higher , I would be a little concerned . Make sure you have the right strings and that they aren't tuned too high . Ask someone at a music store or another musician about it .

A couple things come to mind even though they are obvious.

Have you got the strings seated correctly in the nut and bridge saddles?

Have you replaced with the same size strings or gone to a larger size that doesn't quite sit "in" the nut instead of over the top of the nut string grooves?

Have you performed any neck relief maintenance?

Has the nut shifted or changed its orientation?

I'm having difficulty thinking of a mistake that could have the strings sit noticeably higher above the fretboard.

Did you use strings that are much heavier gauge than what was on there before?
If so, maybe the strings are not settling down into the nut slots properly.

No, you won't hurt your guitar permanently (or at all) by changing the strings wrong.

If the one string isn't at the right height over the fretboard, take a look at the nut and the saddle. Make sure that it's correctly seated in both. If that's not the problem, take a closer look at the saddle. If it's a Fender-style bridge with the action adjustable on each individual string, it's possible that there's something stuck under the saddle, holding it up, or that one of the screws slipped while you had the string off, changing the action.

You probably have not damaged the guitar. However (as was said), you might be making the strings an octave high (too tight), because the high E should not have broken.

And, it will be nearly impossible to keep that broken string in tune.

As a side note (and merely an opinion), I find that those coated strings which last twice as long do not sound anywhere near as good as standard strings. The fact that they normally cost nearly twice as much negates the twice as long factor (unless you just hate changing strings). Also, I find that coating tough to slide smoothly on.

Like some others mentioned, I'd be concerned about being tuned to high which will put to much stress on the neck. I can't think of any reason why the strings would be higher if its tuned properly. One wrap around on the peg won't harm the guitar but it may slip and loosen a bit.

Thank you everyone for the response. I'm glad to hear that the one-wrap around on the high e won't affect the guitar too much.

Regarding the strings sitting higher than before. It is tuned correctly, I am certain of it. The high e broke off not during tuning but when I went back to unwrap the string before wrapping it over again.

I am actully using a ligher guage (light) instead of medium than before. I "think" the strings are seated correctly in the nut and bridge saddles. When I started changing, I simply slipped the string into the nut, pushed the bridge pin in, and everything appeared to be fine. The strings are sitting nicely within the grove up the neck.

One thing that I did notice is that the string sits more and more higher than the body the closer you get away from the neck and to the body. For example, the strings are sitting much higher on the 12th fret than they are on the 1st fret.

the strings are sitting much higher on the 12th fret than they are on the 1st fret.
If it wasn't like that before, then there are really only a couple of possibilities that I can think of:

1) the strings are tighter (which shouldn't be the case with lighter strings that are properly tuned)
2) The bridge saddle is somehow sitting higher than it used to. Unless the saddle was removed, and not re-installed correctly, that's also pretty unlikely.