As explained in the Tips and Notes section:
The basic meaning of this word is on. For example, Ritheann sé ar bhóthar means He runs on a road.
When used with the verb bí, it conveys the idea of obligation to do something. For example, Tá orm rith means I must run. (The literal translation of the phrase would be "It is on me to run".)
Tá ort snámh You must swim
Tá ar Phól éisteacht Paul must listen
Tá orthu siúl They must walk"
What's the difference between "She has to run", "She must run", "She will have to run" or "She is obliged to run"? They all mean basically the same thing, but you might have a preference for one form over another in certain circumstances. It's much the same with Tá ar and Caithfidh (and is gá de and ní mór de and various other phrases). You can probably stick with Caithfidh for your own use, but you need to recognize the other forms when you encounter them in other people's Irish.
No it doesn't say "she wears", and no, it isn't wrong.
The hint for tá ... uirthi suggests that it can mean "she is wearing ..." or "she must", and that uirthi can mean "on her" or "on it".
tá ... uirthi can mean "she is wearing ..." because in English "she has on a skirt", "she has a skirt on" and "she is wearing a skirt" are synonymous - they all mean the same thing - it you have an item of clothing on, you are wearing it, if you are wearing an item of clothing, you have it on.
tá uirthi (verb) means "she must (verb)" or "she has to (verb)" (and any similar constructions used in English).
Tá uirthi rith only fits the 2nd pattern.
Can somebody maybe help me find out how one would classify the parts of speech here? It helps me a lot to put it in order to remember the word orders for different types of sentences. Would it be Verb Preposition Phrase (or Prep + Subject) Secondary Verb? And is "rith" here a verbal noun?