There's not really a need for water dish for any tarantula which are less then 3 inch legspan.
Weekly mist at the side of the enclosure's wall would be ideal, try to maintain 1/3 of the substrate surface slightly moist at all times.
For your second question,
No, not advisable to use sponge/cotton as they might be great breeding ground for bacteria.
A simple shallow water dish with pebbles would suffice.
A recent thread for your reading:http://scorpionforum.darkbb.com/t8955-new-baby-spider-help
By the way, what species is your tarantula?