Learn Finnish: Tuo iäkäs kirjailija on mainekas. - That elderly writer is renowned.

It comes from the noun ikä (age). When adding the -käs suffix, the root undergoes consonant gradation where the k drops out, leaving the stem iä-. Because the suffix implies abundance, someone with "a lot of age" is elderly.

Why does mainekas use the -kas suffix while iäkäs uses -käs?

This follows standard vowel harmony rules. The root word maine contains the back vowel 'a', so it requires the back-vowel suffix -kas. The root word ikä only contains front and neutral vowels (ä, i), so it takes the front-vowel suffix -käs.

How do words ending in -kas or -käs behave when you add case endings, for example in the elative case?

Adjectives ending in -kas or -käs use a strong stem ending in -kkaa- or -kkää- for most cases (though the partitive just adds -ta / -tä). If you wanted to say "about that elderly writer", the elative forms would be tuosta iäkkäästä kirjailijasta.

How does mainekas differ in nuance from the more common word kuuluisa (famous)?

Because mainekas is derived from the noun maine (reputation), it specifically means someone with a highly distinguished or positive reputation. Kuuluisa simply means famous or well-known, which could be for either good or bad reasons.