Learn Turkish: Hastalık geçerli bir mazeret mi?

It comes from the verb geçmek (to pass). By adding the aorist suffix to make geçer (it passes/is current) and then the adjective-forming suffix -li (with/having), we get geçerli, meaning something that is accepted or valid. The opposite would be geçersiz (invalid).

Why is bir placed after the adjective geçerli instead of before it?

In Turkish, the standard word order for 'a/an + adjective + noun' is adjective + bir + noun. So 'a valid excuse' translates to geçerli bir mazeret. If you put bir first, it emphasizes the number 'one' rather than the indefinite article 'a'.

Is hastalık formed from the word hasta?

Yes! The suffix -lık (which follows 4-way vowel harmony) turns adjectives and nouns into abstract nouns, similar to '-ness' or '-hood' in English. Adding it to hasta (sick) creates hastalık (sickness or illness).

How would I use the lesson's grammar of dolayı (due to) if this excuse is accepted?

To say something happened 'due to illness', you take the noun hastalık, add the ablative suffix -tan (following consonant harmony for the 'k'), and add dolayı. For example: Hastalıktan dolayı geç kaldım (I was late due to illness).