I'm not from Sweden but if you could post an approximation of the salary they offered maybe someone who lives there might better answer how you will manage. Good luck!
If you are currently making $7-$10k on average, why would you relocate to an foreign country and make less on a monthly base? Also seems like you are married with 2 children. Relocating your family will be more expensive than your first thought because most probably you will have to cover their visa expenses. You all need to get familiar with "aliens".
If I were you I would offer remote service to that company but would not relocate there. If they accept it's a win win. Other wise you can continue freelance.
That sounds about average for Stockholm I believe.
The salary will obviously be lower than freelancing rates since there are less taxes and costs to consider. You will have to make a spreadsheet or something to see what would end up being most profitable.
Housing is rather expensive in Stockholm at the moment, and rentals are hard to come by unless you want to pay ~10k SEK for a small one bedroom flat.
I can confirm that that is a realistic average for a frontend guy. Keep in mind too that most larger companies will offer you a salary that is significantly below the median they offer Swedish applicants because they believe they have a leg up on you. You are in a worse negotiating position and they expect you to be used to work for a lower wage.