Are the ATC transponder Mode A and Mode C still in use (according to ICAO or FAA rules)?
Yes. According to Title 14 CFR §91.215 ATC transponder and altitude reporting equipment and use.
(a) All airspace: U.S.-registered civil aircraft. For operations not conducted under part 121 or 135 of this chapter, ATC transponder equipment installed must meet the performance and environmental requirements of any class of TSO-C74b (Mode A) or any class of TSO-C74c (Mode A with altitude reporting capability) as appropriate, or the appropriate class of TSO-C112 (Mode S).
the use of Mode A or Mode C (e.g.: Mode A with altitude reporting capability) is acceptable for flight under Part 91. That covers all of general aviation (GA), of which a large percentage use Mode A/C transponders.
With the upcoming ADS-B mandate, many GA operators are switching to Mode S to get the 1090 MHz ES ADS-B. But others are keeping the Mode A/C and adding a 978 MHz UAT to meet the ADS-B requirement. In any case, there are still a large number of Mode A/C transponders in use and likely will be the case for quite a while.
Mode S is optional and only required for operations under Parts 121 (Air Carriers) and 135 (Air Taxi).
For Part 121, a Mode S transponder is required by §121.356 Collision avoidance system.
For Part 135, Mode S is required by §135.143 General requirements.