1- if you come as a tourist, you can stay for 3 months, but all costs you have to pay for yourself. A good option is to get the European health insurance card, so you can get some of the Kela services for free (get the info from the DHSS)
2- after 3 months you have to register with the police. They can require a prove of income. I know of cases where teleworking was not accepted, but it is the judgement of the local officer.
3- if you want to avoid all the fuss, marry him.