Translation:The naughty cats bite the old grandma.
How informal is Finland when it comes to giving strangers nicknames? Is mummoa interchangeable with old lady and grandma? I would never call an older woman "grannie" if she wasn't part of my family, it would be considered rude on my part. Is eukko acceptable instead if I want to show respect?
Eukko is not acceptable and it's a very rude word. It probably has been used more but nowadays it's used mostly in stories like fairy tales and such.
Mummo is in my opinion a pretty neutral term and I use it quite a lot. You don't have to be a grandma to be a mummo, just an old lady. Many use it also to refer to their own grandma.
I use the word mummi to speak about my grandma. It's used a lot but not usually to refer to all old women or grandmas in general, it's a word usually for your own grandma (or someone else's if they use that word for their grandma).
Then there's isoäiti. It means only grandmas (women who have grandchildren) and it's more formal.
And then there are tons of other names but these are probably the most used. So in conclusion, just go with "mummo" if you talk about grandmas or old ladies in general. It's not the same though as in English and I wouldn't maybe call someone mummo to their face if they were a stranger. Täti (aunt) is also used of middle aged and older women but still wouldn't say that to a stranger and people don't really use it that often.
Oh gosh, yeah, don't use eukko, not even about a third person. I agree with all of Anna's other points, as well.
I think the best way to refer to an older woman you don't know, if wanting to catch her attention for example because she dropped something in the street, is rouva (Mrs) -- regardless of her marital status. Of course, if you can do it just by saying Hei or Anteeksi, even better!
If you talk about an elderly woman you don't know to a third person, you could say rouva, nainen, or vanhempi rouva/nainen. Illogically (but this is the case in many languages) vanhempi (older) is actually less strong than vanha (old). Mummo is still a bit less respectful and perhaps overly familiar than these terms.
Do let us know if you have questions about specific situations!
Yes, I agree! I totally forgot about rouva when typing this last night. That's a very polite way to address an older woman. You also might want to use 2nd person plural if you want to be very polite. It's used less nowadays but you can still use it sometimes. Vanhempi nainen is a very neutral term and that's used a lot these days. Overall, informality is not a big deal and overpoliteness sounds a bit funny in Finnish.
"Older" being less extreme than "old" is not illogical at all. Comparative adjectives are not extreme adjectives. "Older" doesn't mean "very old". "Older" is relative to something else; "old" is not and is more of an absolute judgement.
No one would call a 3-year-old child "old" without qualification, but in relation to younger siblings, they are older.
Saying "an old lady" is saying that she's old. Saying an "older lady" is not saying she's old in any absolute sense but instead in a sense relative to someone or something else.
"He likes older women," can mean that he likes women who are older than he is. If he's 23, a 30-year-old woman is an "older woman". "He likes old women," would surely mean that a 30-year-old woman is too young for him because she is not "old" in an unqualified, absolute sense. (Of course, even unqualified "old" is still relative to what you're talking about - an old cat is not as old as an old planet!)
She is the object of the biting, but puraista is not one of those verbs that almost always lead to the partitive. She also probably (and hopefully) doesn't go through a significant change because of this. If this were a nightmare scenario and the naughty cats would bite and kill her, the object would be the "complete" object: Tuhmat kissat puraisevat vanhan mummon kuoliaaksi (to death). You'd still need some word there to explain how come you used the complete object. But you don't always, you can say "Ammuin rosvoa" = I shot at the bandit or "Ammuin rosvon" = I shot and killed the bandit. I guess with shooting, the possibility of dying is somewhat more obvious than with biting...