Learn Dutch: Haar neus is rood.

You just place it right before the adjective: Haar neus is wat rood. Here, wat doesn't mean 'what'; it acts as an adverb meaning 'a little' or 'somewhat'.

Why does the adjective not get an '-e' at the end, like in rode?

When an adjective comes after the noun it describes (usually connected by a verb like is), it never gets an -e ending. You would only use rode if it was placed directly in front of the noun, like in haar rode neus (her red nose).

If I change this to Haar neus is wat rood, could I also just say Haar neus is een beetje rood?

Yes, absolutely! Een beetje and adverbial wat are completely interchangeable here. Wat is just a shorter, very natural-sounding option that Dutch speakers use constantly to soften a statement.

Doesn't haar also mean 'hair'?

Yes, it can! Haar is the possessive pronoun for 'her', but het haar is the noun for 'the hair'. You have to rely on context to tell them apart, though here it is clearly a pronoun indicating whose nose it is.