Is it the Symbol D visa that you're currently on? If so, this can be extended as long as it's done 30 days before its expiry date. See here for more information. The page also goes into more details of each visa subtype for D visas so that may be worth checking out.
You can have a look at this picture. So what you'd be looking for is the part labeled "visa symbol" next to the top right hand corner of the visa photograph of you. The D visa is usually what's used for volunteers, but maybe you have the C visa? See this page for more info on that, but I think your visa should be a D.
You might want to contact your local embassy or consulate to be safe if you're still uncertain - they'll be able to give you a definitive answer.
For staying in Romania longer than 90 days, you need a temporary residence permit. Do you already have one?
It looks like you can only apply for permanent residency after living there for six years, but if you get a work permit (which you'd need anyway to work in Romania, so I assume you have one?), you can apply for a long-stay visa and a stay permit. See here and here for more info.
If the company you're working for is a Romanian company, you should be able to apply for a work permit even if they're based online. Again, to be sure, you should check with your local embassy just in case.