2) How much does a decent ( 800-900 sq feet apartment ) place in a clean and safe neighborhood cost ?
You should start thinking in metric, they don't use feet and inches in Poland :) so you're looking for something around 70 square metres. Here's a link to listings for apartments in Gdansk:
domiporta.pl/mieszkanie/wynajme/pomorskie/gdansk
In addition to the rent stated there is usually an extra service charge for cold water, rubbish collection etc called a Czynsz and that's generally about another 500 a month. Sometimes, depending on the type of building this will include central heating and hot water but not always. The ad will usually list the service charge separately. Your electricity bill will be about 100 a month and you might have gas too. Internet, about 60 a month.
As for neighbourhoods, well because of its history and social housing, Polish neighbourhoods tend to be very socially mixed. You will find local authority blocks of apartments right beside expensive luxury private blocks so there's really no such thing as a completely upmarket/exclusive neighbourhood or an indeed a bad part of town. Pretty much everywhere is 'clean and safe'.
ow much does decent vegetarian eating for 2 ppl ( including once/twice a week eating out ) cost ?
It's not easy to find vegetarian options in Poland. It's still very much a meat eating culture so your choice will be somewhat limited if you're eating out. You can pay pretty much anything you like for a meal. A cheapish but acceptable main course will cost around 20 pln up to about 100 for an upmarket option.
By the way, you should consider a smaller apartment if it will save you some money. For two people, 50sq metres is plenty of space and could be a thousand a month saved, far better to have that 12 grand in YOUR pocket than in a landlord's :))