Well, with regards to a couple of your points I'd expect the whole thing to be paid for (flights, hotel, transport on the ground etc). The only thing that differs from company to company is how strict they are for food/drink -- if there's a limit on anything. Some places I've heard you can only have one alcoholic beverage with dinner, nothing else. Other you get £15 for lunch, £30 for dinner. My place, no limits whatsoever
So if I go on a trip I have my business credit card -- everything goes on that. Flights, hotel, food, train to/from the airport, taxis etc. -- even the random coffees, snacks if I'm out and about. Then a few drinks perhaps
I wouldn't personally pay a penny on anything and my place is absolutely fine with that. So I'd check with HR/whoever, and maybe check with others that have travelled a lot on the company how stringent they worry about expenses. It's good to know before you go as you don't want to rock up to a posh restaurant and end up getting stung for the £100 bottle of wine because you can't expense it.
It's a bit hard to do sightseeing on an evening if you're going to be solidly in work for the daytime. If you've got two nights out planned then you're halfway there. I wouldn't worry too much. I'd also imagine that your place has business travel insurance in place, but best to check.
EDIT: I had 3/4s of a day free in Paris a couple of weeks ago. I actually did an open-top bus tour for what it's worth. Quite fun actually and if you're short on time you get to tick off the main sights. Maybe there's one in Bucharest?