Yes, for me it is clear, when reading your sentence, that he/she said, that he/she already has been shopping/to the market. That would be a repeating though. Other prominent languages like English, French, German, don't have that afaik, so for most of us it is a very new thing to learn :)
That seems to be like the short answers in English to me - I'm not quite sure what happens there. There it could be possible, not using them - If I have some time the next days I may look it up in my big red grammar book :) Just remind me, if I forgot, that's ok!
I was talking about "normal" sentences, like hmwlW2 gave an example...
You can leave it out if you want.
Olla (to be) conjugation in present tense:
The ending already tells who is the subject, so there is no need to add the subject pronoun unless you want to emphasize it.
Btw, there are very few irregular verbs in Finnish, but this is one of them.