Actually for his guitar the deeper it is into the nut the better. I believe you have a guitar that employs a zero fret. Where instead of having the nut be the contact point at the head stock another fret is used, this is mainly used on cheaper guitars since the cost of the labor involved in making a nut with slots carefully filed to the correct height is greater than the labor required to install a zero fret.
And your bridge looks very peculiar, it seems like your guitar uses an archtop style floating bridge. So there should be an actual bridge piece that spaces the strings down at the guitar body's end, it seems to be missing from your pictures hence the action becoming way too low. Look around for it, it's quite hard to miss, it might have fallen off when you were restringing. You must have taken all the strings off at once to restring where instead you should restring one by one especially with these guitars unless you need to clean the guitar or install pickups. I suggest you bring it to an experienced person to do it for you as you can't just pop the floating bridge anywhere you like, it needs to be intonated.