You will unless they are closely related enough to give you citizenship rights. Nationality law is completely different from country to country within Europe, though.
Depends how she got citizenship. If she got it through bloodline, maybe you're eligible too. If not, you can come as a tourist. Probably 3-6 months, depending on country. May be eligible for cheaper visa if you'd supply an invite from her.
Keep in mind that major non-EU countries (US/CA/AU/etc) don't need visa for tourist stuff. You'd get a stamp at airport for x months and that's it.
It may also matter how much money you make, whether you're married to someone who is taking benefits, and so on. Even for the countries I've lived in, it's been a mess to find out.
And when you do find out, sometimes you find there's some loophole. For instance, if you try to bring a non-EU spouse to Denmark, it's hard if you have low income. However, Sweden is right across the water from Denmark, and there's a bridge connecting them. And the Swedes have a different national law. Also there's EU law, which is another way to get in.
Oh, and the UK voted to leave the EU, so there will be even more legal spaghetti.
Try to narrow down what countries you're interested in, and find someone who can help you.
Unless you're a citizen of a EU country you need a residence permit (people call this "visa", but technically they are 2 different things) to work in a EU country. As a US citizen, you can visit for 90 days without a visa, but after that you'd need something.
AFAIK if you have a grandparent who is an Irish citizen and was born there, you can get an Irish citizenship. That is automatically an EU citizenship, and if you have that you can live and work anywhere in the EU without needing any sort of permit.
But even if you have this Irish grandparent, you can't just move into a EU country and live there BEFORE getting an official document saying you are now an Irish citizen.
Depends. Europe is not a single country, it's impossible to answer this question. (unless in the case where you can in every country, but I wouldn't know).