Export your finished design as a jpeg or png, at the size you want it to be displayed. Adding images in email signatures is fine as long as you realise not everyone will see them. Many people, including me, switch off loading images in emails since it's an annoyance. Also, some people block emails with images in them, because they look spammy, or it may be company policy to block such emails, for security reasons.
Your second question depends on the licence for the font. You should check the licence of each font you want to use. The licence should tell you how you can use it. Generally speaking, if you have a licence (whether paid or free) that allows you to use a font, you can use it for anything you want. However some free font licences may prohibit commercial use of a font. Obviously, a public school is not a commercial company, but there could still be legal issues. Again, check the licence of the font for specific details.
If you want fonts that can be used freely for just about anything, have a look at SIL OFL fonts, or even Google Fonts.
The copyright of a font covers the re-distribution of the font file itself, or the making of derivative fonts, not the normal use of the font. Otherwise nobody would be able to use a font for anything.