AVR Embedded Webserver + Time
Periodically check the time (using NTP or such) and update the internal RTC with the right time


Shouldn't be a problem although the Atmega328 doesn't have an internal RTC.  It has a timer but that will fill up.  You will need something like a DS1307 if you need an RTC.


  Should I go with ENC28J60 or Wiznet5100 or higher chip.


The Wiznet would be preferable although you can opt for better security with their better stack or go with the ENC28J60 and get a strong community backing.


  Any suggestions for the web server? Ideally I would like it to look polished with images and such.


While it won't make the greatest web-server, people have done this. I have been curious about it myself.  Check out this very promising link

You won't be able to do too much with the server but you can do some basic GPIO, SPI, and I2C stuff.


  For time part, should I go with NTP or the simples (but older) alternatives such as daytime or time?


This is entirely up to you

Good luck!