Learn Spanish (Latin America): Aunque hay barro, ella continúa.

Because we are stating a known fact. The speaker knows for sure that there is mud right now. When aunque (although or even though) introduces a definite reality, we use the regular present tense, which is also called the indicative mood.

Could I use está instead of hay to say 'there is' in this sentence?

No, you must use hay. Hay is always used to say 'there is' or 'there are' when stating that something generally exists, like the mud in this sentence. You only use forms of estar (like está) to talk about the location or state of a specific thing you have already mentioned.

Why is there an accent mark on the 'u' in continúa?

The accent mark tells you to put the spoken stress on the u and breaks the 'u' and 'a' into two separate syllables (con-ti-nú-a). Without the accent, the rules of Spanish pronunciation would blend the 'ua' together into a single syllable.

Can aunque be translated as 'even though' instead of 'although'?

Yes! Aunque translates to both 'although' and 'even though'. Saying 'Even though there is mud, she continues' is perfectly correct and means the exact same thing here.