Sijambo is not really a verb. It is the noun 'jambo' which means problems / incidents and the first person sg negative prefix 'si' - I don't have
In other words 'sijambo' means 'I have no problems' meaning 'I am fine'.
That explains also the the question:
Hujambo ?-You (sg) don't have problems? - Are you alright?
Hamjambo? - You (pl) don't have problems ? - Are you (pl) alright?
and the answer to the last one:
- Hatujambo. We don't have problems - We are fine
Jambo is singular, so not problems/incidents in the plural but simply problem ... it actually just means "thing", as in an abstract thing (not a physical object), so it's kind of like "matter" or "affair".
Sijambo isn't I am so much as I am fine. I suspect the sense is something like '(I am) not bad', said in response to 'How are things?'. Or as the Scots would say, 'Nae bad'.
That's a good question!
It's not true with "have". The na is basically just "with". The infinitive is kuwa na, literally "to be with" (not kuna). You'll also notice that it doesn't have the tense prefix either. Essentially, it's not a verb, it's a preposition that takes verbal prefixes for the present tense.
Si means is not (I do not) and Na means have. So Sina would be I don't have