I think a lot of FIRE discussion is US-centric, although this board clearly has an international contingent as well.
The thinking behind an emergency fund is that one may, from time to time, have large unexpected expenses. These can be met either via an emergency fund, credit/debt/borrowing, or selling of assets. You're right in thinking that an emergency fund has the opportunity cost of not being invested. Some rely on credit/debt/borrowing, but that comes with interest expenses. Selling of assets can work, but in the US there may be capital gains taxes due if selling appreciated investments, which are taxed at ordinary income tax rates if the investment is held less than one year. Selling of, and removal of assets from retirement or other preferenced accounts in the US can be subject to both taxes and penalties. So an emergency fund, depending on your perspective on these other options, may be the best of several not great options.
The usual things I hear mentioned in regards to large unexpected expenses is a job loss or a medical event. And you're right, if those are covered by the social safety net of your government and you believe that to be adequate for your needs, then those emergencies are covered. Other possibilities for emergencies might include the opportunity to buy some underpriced item to resell (such as a car), an actual sudden problem with your car that requires immediate replacement (unlikely, and in Europe you might not even rely on a car - the US is much more car-centric as I'm sure you know), or a severe problem with your house (although there can be insurance for that as well).
I had an emergency fund while I was working mainly to ensure that I could continue to pay child support in the case of job loss. I kept three or four months of expenses and invested anything above that. I never actually used it for an emergency, and now that I'm FIREd I don't really have an emergency fund; if I had something happen I would just pull from my FIRE stash.
I also happen to think that some decent percentage of time emergencies can be minimized, avoided, or planned for. I'm 51 and actually never used my emergency fund. Maybe I lead a boring life.
No advice on ETFs in Europe. I think investing, investment companies, and investment rules are very different between here and there and I have no knowledge or experience in European investing. I'd suggest low-cost, broadly diversified, high quality investments, and LTBH. Personally I am and have been highly invested in equities my entire life, but that requires a certain investing mindset and risk tolerance that not everyone has. And it may not be appropriate for your life situation / goals / timeframe /etc.
In the US we have Social Security, which one can start between age 62 and 70 and you get a monthly check for life. The amount of the check is generally based on how much you (or in some cases your spouse or ex-spouse) earned, with lower income people getting a larger replacement percentage than higher earners. The actuarial soundness of the system is questionable, and people therefore have various opinions on how much one should rely on it. You're right in that the 4% rule generally doesn't account for the fact that Social Security will partially supply income needs at some point during a person's retirement. Some people choose not to account for Social Security and treat it as extra. Others choose to count on it to some degree, either by subtracting it from the spending need starting at SS age or by adding an NPV of those future payments to their FIRE stash.
One minor correction - the 4% rule is not really a mathematically based rule - it's based on historical studies of how people would have done in the past if they had $X invested in a portfolio with a particular asset allocation and withdrew 4% of that every year adjusting that amount for inflation over 30 years. Notably, most 4% rule research (as far as I'm aware) is based on the performance of US investments and US inflation. Your country and your investments may have different historical returns and inflation rates, which is something you may want to consider when evaluating your plan.