Is it okay to connect a European 240V transformer to a US 240V outlet?
Yes, you can connect that to a US 220-240 V outlet. Just connect to the two hot lines and connect nothing to the neutral. However, there are normally only 30 to 60 amp 240 volt outlets available in US residences. They are dedicated to clothes dryer cooktop and oven use. To add a 240 volt circuit requires two adjacent 120 volt circuit breaker spaces in the distribution box. You will need a 15 or 20 amp double breaker whichever is the smallest available. You can connect additional 120 volt circuits to the same breaker, but they must not be circuits like bathroom and kitchen outlets that can not be mixed with outlets elsewhere.
It is not a good idea to connect to a higher current circuit because the higher rated circuit breaker might not prevent a fire if you have a short-circuit in the charger or in the charger cord. If you want to do that you should have smaller fuses near the outlet. I don't know if that is permitted by code.
You should use a US receptacle and plug rated for 240 volts and the fuse or circuit breaker current rating. You probably will not find a suitable converter for a 240 volt US receptacle.
There could be a difference in performance related to the actual voltage in the previous vs. present locations. However the transformer will be operating within the specified voltage and frequency range, so that is nothing to be concerned about. If the new voltage is higher, the charging current may be a little higher and the charging time a little quicker and the opposite if the voltage is lower. I assume an old charger may not have any voltage regulation. The operating temperature may be lower or higher.
A 15 or 20 amp 240 volt receptacle installed for a window air conditioner is an excellent choice for this application.
If the device has a Schuko plug or an Europlug, you should be able to use it on a split-phase system. The plug is unpolarized anyway, so the device has to be able to accept live voltage on any of the two pins. The mains power switches in Schuko devices always switch both live and neutral for this reason.
You should just make sure to use an adapter with an earthing pin if the charger uses a Schuko plug. Operating a Schuko device without earthing might be quite dangerous.
Sure, there's no problem. You see common 120V receptacles all over houses in North America. You can do exactly the same thing with 240V circuits.
The normal 120V plugs are called NEMA 5-15 plugs, and they go into either NEMA 5-15 or 5-20 sockets. The Electrical Code says the breaker must be 15A or 20A, and the socket ampacity must match the breaker, except 5-15 sockets are OK on a 20A circuit if there's more than one socket.
The same exact rules apply to 240V circuits (no neutral) except you use NEMA 6-15 and 6-20. The sockets are almost identical but are keyed not to fit NEMA 5.
You could take a 120V circuit and just convert it to 240V by changing all the sockets to NEMA 6 and landing its 2 conductors on a 240V breaker. However, Code requires certain mandatory sockets in homes, and those mandatory ones must be 120V. Other than that, fit all the NEMA 6-15 or 6-20 sockets you desire. It's no more difficult than installing a 120V circuit. You cable it with the normal /2 cable (twin and earth) and you tape the white wire to indicate it is a hot.
Now, one person talked about making a 240V circuit that serves both 120V and 240V loads. That's allowed but is a little awkward. It needs the more special /3 cable with two hots and a neutral. From the 120v circuit's perspective, it is a multi-wire branch circuit with some funny rules. GFCI/AFCI, if needed, wont be any harder than on a straight 240V circuit, but will probably end up needing to be in the breaker panel.