Couple moving to Vienna in July

Hi!

We are a couple from Bucharest, Romania and we're planning on moving to Vienna in July. We are freelancers and we'd like to live there indefinitely so we'll be applying for a residence permit.

We are very stressed out, there's a lot of unanswered questions and the more we look online, the more we get confused.
So here goes:
1. How long does it take to find an apartment? We have an open collaboration with an agent and we've asked him this and he always says he can't say exactly. We are very limited in the time that we have, because we have to find a short term accommodation, so we've accounted for 1 month. But, does it really take that long?? 1 month to find an apartment? We're hoping we're just being paranoid and it will take less or at least not more than one month.
2. Do all apartments fall under the tenant's protection jurisdiction? That set of laws meant to protect the tenant...If not, should we just look and sign contracts that are regulated by those laws?
3. After we find an apartment and register our address, we'd like to start the residency applying process and also pay taxes and social contributions there. So, keeping into account we are freelancers, we are from EU, what do we have to do first?
Do we first apply for the residency and then apply as freelancers and pay taxes or the other way around?
Also, what's the deal with medical insurance, because I read it's a must for residency, so if we don't apply as freelancer first and pay social security contribution, does that mean we'll have to get private insurance? How much is that?
Assuming we'll have to get private insurance for a while until our contributions kick in, since we are in EU does our home country health insurance count there and we'll be exempt from getting private insurance?

Please help!! Thank you!