1.Who looks after our flowers while we are on holiday?
looks is present tense. this is something someone might ask while they currently away on holiday.
2.Who is looking after our flowers while we are on holiday?
same as above
3.Who will look after our flowers while we are on holiday?
future tense. this is what you would ask before you leave, your holiday vacation is still in the future.
4.Who will be looking after our flowers while we are on holiday?
this is future progressive. it is better suited to something ongoing and future.
5.Who is going to look after our flowers while we are on holiday?
this is okay, just not as good as 3.
6.Who is going to be looking after our flowers while we are on holiday?
not grammatically wrong, but too indirect.
also, saying "on holiday" is not really normal. a holiday is non-specific period of time. you might say "during holidays?" or if since you're probably away from your flowers on some kind of trip, "on vacation?" would be most common.