1) Taxes on Bitcoin sales do not get automatically deducted in any European country, you will have to file tax reports yourself.
So the banks don't rat you out as the U.S. banks do?
What I mean is that I'm not aware of any broker that offers automatic taxation on Bitcoin trades (brokers and banks usually automatically deduct capital gains tax in case of funds and bonds for example).
This means you'll have to manually file tax reports (or have a tax advisor help you) and then pay taxes according to whatever tax rate applies to you.
Whether banks will report "shady" incoming transactions from crypto-exchanges or take action on it will depend on the bank. Usually has nothing to do with taxation though, rather with worries about money laundering and such.
2) Taxes vary wildly between EU member states. Not just in regards to crypto. Be aware that the tax-free-after-1-year-of-holding rule only applies to a handful of countries (eg. Germany, Austria,... not sure where else). Selling before the 1 year holding is usually taxable as income tax (which also varies wildly between member states), selling after the holding period is usually taxable as capital gains tax (or not at all, if you're lucky). Either way, look closely into how each country handles taxation.
Yes, I know. Assuming it's one of the countries that do that, how would reporting work? I don't think you'd have to report then, right? Why do you say "if you're lucky?"
With "lucky" i mean if you're lucky to be tax resident of one of the countries that don't tax Bitcoin gains after the 1 year holding period

If you're a tax resident of Germany or Austria and are not required to pay taxes due to the 1 year holding period... you're also not required to report anything. It supposedly may prevent you some paperwork a couple years down, but in principle you're not obliged to report those gains.
If you have to pay income tax because you sold prior to the 1 year holding period, you need to fill out the respective income tax forms and the ministry of finance will tell you how much tax you owe. Of course you could just... not report it... but then if they find out you'll be on the hook for tax evasion.
I can only talk about Austria and Germany though, as I'm a tax resident of the former and have some connections to the latter. Other European countries may handle this differently.
3) In which country you bought the Bitcoin originally doesn't matter. You should be able to prove holding time though. Arguably referring to the blockchain may suffice, but as far as I know there has been no precedence on how to report holding time correctly so far -- and you can also expect that what counts as proof for holding time will vary wildly between EU member states.
Well, has anyone reported time so far? Has it worked? What's the point of guidance on these things if a tax payer can't report it properly or it won't get accepted?
¯\_(ツ)_/¯
I'm afraid we're on uncharted territory, my friend.
4) You pay tax in the country you reside in, assuming you have already taken full residency and have become a tax resident. At which point you become a tax resident depends on the country you move to and likely also where you are originally emigrating from (ie. immigrants from non-EU member states have to follow different rules than immigrants from EU member states. I wouldn't be surprised if there's a set of rules specifically for immigrants from the US). Prepare for paperwork as you likely won't become a tax resident automatically, but also will need to become a tax resident at one point to be allowed to legally stay there for a prolonged period of time -- assuming you don't have citizenship already.
That sounds insanely complicated... I have no idea how to go about this.
It's simple in that the basic principle is that you have to pay taxes in the country that you live in. It's complicated in that each European country handles immigration differently. If you pick one country and really look into it, the process should be rather straightforward.
Wow, super useful. Ok. Then I guess when I follow my friend's path of handing in my passport, it'll be over 1 year of holding and then I can't legally be a resident of the previous country, so... I'll end up withdrawing in a country that has the past-1-year-holding policy. Thanks for the info! Really appreciated.