UART frequency in microcontrollers
1) Why the frequency should be 16 times the baud rate?


As Chris has already pointed out in the comments, there is no clock signal, RS-232 UARTs are asynchronous systems. This means we have no idea when incoming data will start and it's a fairly safe bet that the transmitter clock will be running at a slightly different rate to the receivers. In order to cope with his the UART oversamples the incoming data so that it can come up with a reasonable measure of where the transitions in the data are.
In order to oversample you need a clock running significantly faster than the data rate.


  2) Can you give me an example of any Microcontroller where we set the
  UART frequency?


Virtually all of them will contain various registers to set this. e.g. on the LPC1768 there is the Fractional Divider Register to set the clock pre-scaler to the UART logic.
Generally setting the clock frequency for the UART logic is also setting the baud rate for the output and so it may not be obvious that you're setting a clock to 16 times the baud rate.


  3) What the purpose of setting clock frequency to UART if we have
  timers? Like we generate baud rate in simple 89c52 through Timer 1 and
  we don't set any frequency.


So that you can run each UART at a different baud rate without using up all of your timers?


  4) What's the idea of setting the baud rate with timer like we do in
  89c5x. How it is linked to UART frequency?


No idea, not used the part but that seems a little odd.


  5) RS-232 is recommended standard 232 and UART is physical. Hardware
  does parallel-serial and serial-parallel conversions. Am I right?


Yes, the UART has an 8 bit transmit and receive buffer, the processor only deals with bytes of data, the UART hardware handles converting to serial and back.


  6) Does UART contain a counter, so that after the start bit it waits
  for 8 cycles and then after each 16 cycles, the samples are taken from
  RX line to detect the bit?


It contains a state machine that tracks the current point in the transmit/receive cycle. This state machine will contain both counters and logic to detect transitions, start and stop bits and generate/check parity bits. Unless you want to design your own UART you don't need to worry about what exactly is going on inside that state machine. If you do want to design your own UART then that's a whole new subject.