3 nights will give you a good chance to see the northern lights if that's what you want to see.
A lot of places do a free second trip if you don't see them the first night, so book early on so you have a second chance (we made this mistake with the whale watching cruise - we saw dolphins, but no whales apart from a very fleeting glance so qualified for the free trip, but couldn't take them up on the offer as it was the day before we came home).
But 3 nights will only scratch the surface of Reykjavik, let alone Iceland, so you'll want to go back unless you decide it's not your thing.
As for food/budgeting, it's a bit 'how long is a piece of string'. It's the kind of place where you need to spend money to make the most of it, and most things are quite expensive, but there are ways to limit the cost such as buying the city pass and spending a day or two visiting all the included museums and swimming pools.
Eating out at lunchtime isn't too bad - you could buy hot dogs and pizza by the slice fairly reasonably and sandwiches, coffee and cake in coffee shops was actually cheaper than going somewhere like Costa in the UK and better quality as all fresh and home made.
If you self cater that's obviously cheaper than eating out, but supermarket food was usually 50-100% more than the UK. Some evening meals were comparable to UK but alcohol was more, although it was cheaper during happy hours, which were often 2-3 hours long, eg 4-7 pm but you could spend a lot on a meal out - eg steak or fish and wine would probably have been about £100 pp for a 2 course meal and a shared bottle of wine in even a mid range restaurant.
The most popular tours like the Golden Circle or a whale watching tour was about £60-80 for an adult.
We really liked it and will definitely go back as there's so much that we didn't do.