In The Cricket Playing Without Wicket keeper Is Possible?

It is Legal to play without a wicketkeeper in a cricket match.

It's possible to play a cricket match without wicketkeeper, But the game will be remembered for one of the strangest field placings seen in English cricket for years.

There is no rule stating a team must play a wicket-keeper. On 5 June 2015 during a T20 Blast game between the Worcestershire Rapids and therefore the Northamptonshire Steelbacks, Worcestershire chose to not play a wicket-keeper within the 16th over of the match. Their keeper, Ben Cox, became an additional fielder at fly slip while spinner Moeen Ali bowled. The umpires consulted with one another and agreed that there was nothing within the rules to stop it from happening.

The only restriction upon the team is going to be that if the wicketkeeper is playing as a fielder, he/she won't be allowed to use any equipment like pads or gloves.

According to ICC law 40.1 :

If by the wicket-keeper’s actions and positioning when the ball comes into play it's apparent to the umpires that he/she won't be ready to perform the traditional duties of a wicket-keeper, he/she shall forfeit this right and also the proper to be recognized as a wicket-keeper.

Can a Wicket-keeper bowl during a match?

A wicket-keeper can definitely bowl during a match as there no restrictions on bowling without fielding one over prior but someone from the bowling team must keep the wickets instead of him.

We have seen Mahendra Singh Dhoni bowling repeatedly though he's a wicket-keeper without fielding an over before Virat Kohli or Rahul Dravid kept wickets while he was bowling.