Do airport border officials know which flight you came on?

There is much less automated exchange of data than what many people assume, so the answer to your question as asked - "do they know" - is probably no. If you however ask: "can they easily find out", you must assume that the answer is yes.

In most cases, the airline is obligated to file a flight manifest with passenger data to the relevant aviation authorities and in some cases even to the immigration authorities in advance. For example, US Customs and Border Protection will always in advance be informed by the airlines about which passengers are coming and from where. If the immigration authorities do not have the information in advance, you must assume that they can get the data from the aviation authorities on demand. These records may or may not include data regarding previous connecting flights. If you however fly from A to B on one ticket from one airline and later from B to C with a separate ticket from another airline, the last airline won't know that you originally came from A and you can safely assume that there are no traces of that flight in the last flight manifest.

If a border officer is in doubt, there are however many other ways to more or less easily find out where you have been. It is nearly impossible to exist nowadays without leaving electronic traces of your whereabouts and especially if you enter your home country, border officials may very well be able to get access to this information if they require so. They don't have it when you present yourself at the immigration desk at the airport, but if you are questioned, something does not add up in your story and they find a reason to investigate further, I would consider it more than likely that the immigration authorities have legal means to request data from e.g. your bank or your cell phone provider, which show where you have been. In many cases, disclosing information about you can even be found in social networks without much effort. Posting a picture of yourself on the beach in country A and a few days later to tell an immigration officer that you haven't been there can easily go wrong.

You must also remember, that you in this case probably not only are breaking immigration laws by lying to a border official, but are (also) violating disease control legislation. The penalties for doing so can be surprising, especially if it turns out that you actually are infected with corona (which currently is an unlikely, but not unrealistic possibility when coming from Europe), lie to circumvent quarantine requirements and then with bad luck manage to infect other people.

Yes, for many countries, airlines must send APIS data to the destination country containing a complete passenger manifest. The border authorities of such countries will know exactly who was on which inbound flight.

For example, if you enter the US and you use an APC kiosk, the kiosk will ask you what flight you came in on and check this against APIS data. If there is a mismatch you'll get the dreaded X on your receipt and have an uncomfortable conversation with a CBP officer. You should assume that other countries do similar (manual or automated) checks.

The border officials will ask you where are you coming from. Do they have this information already and they just test you?

They could be asking you or they could be testing you, you never know. If you answer with hesitation/uncertainty, I don't think it is hard for them at all to check your passport and your country of origin. Lying is not a good idea, if travel is not allowed from your country, don't travel.

Also what happens if you have a stopover in another country. Will they know that you actually come from another country and not the stopover one?

Flight stopovers are usually linked on a ticket, so immigration will most likely see where you've come from. If you booked two flights on a seperate ticket, they may investigate further and wonder how you went from the stopover country to the destination country when you do not live/reside in the stopover country.

I wouldn't advise travelling, after all doing this could cause you more trouble in the future than it's worth.

I don't know the answer for Romania. But for Canada the answer is "yes", so a similar system could be in place.

In Canada you're required to fill out a form on arrival with your name, passport number etc, and one detail is the flight number you came in on. Recently some airports switched to electronic terminals. You scan your passport and it pre-populates most of the form including the flight number you arrived on. Therefore this information is clearly available and associated to your passport.