Do all 11 fielders need to be in the field?
Yes,if anyone is outside the rope it is a no ball
Restrictions on field placement
Fielders may be placed anywhere on the field, subject to the following rules. At the time the ball is bowled:
No fielder may be standing on or with any part of his body over the pitch (the central strip of the playing area between the wickets). If his body casts a shadow over the pitch the shadow must not move until after the batsman has played (or had the opportunity to play) at the ball.
There may be no more than two fielders, other than the wicket-keeper, standing in the quadrant of the field behind square leg. See Bodyline for details on one reason this rule exists.
In some one-day matches:
During designated overs of an innings (see Powerplay (cricket)), there may be no more than two fielders standing outside an oval line marked on the field, being semicircles centred on the middle stump of each wicket of radius 30 yards, joined by straight lines parallel to the pitch. This is known as the fielding circle. In addition, during these overs there must be two fielders (other than the wicket-keeper) in designated "close catching" positions.
For the remainder of the innings there may be no more than four fielders standing outside the fielding circle.
In addition no fielder may stand behind directly behind the wicket keeper. Fielders may be placed anywhere on the field other than the pitch or behind the wicket keeper.
The restriction for one-day cricket is designed to prevent the fielding team from setting extremely defensive fields and concentrating solely on preventing the batting team from scoring runs.
What action is taken if the rules are violated
If any of these rules is violated, an umpire will call the delivery a no ball. Additionally a player may not make any significant movement after the ball comes into play and before the ball reaches the striker. If this happens, an umpire will call and signal 'dead ball'. For close fielders anything other than minor adjustments to stance or position in relation to the striker is significant. In the outfield, fielders may move in towards the striker or striker's wicket; indeed, they usually do. However, anything other than slight movement off line or away from the striker is to be considered significant.
I didn't find any link for your another question
Can't the fielding captain argue that we do not the 11th man on the
field, for whatever reasons?
I have not heard about such scenario also if the captain argues with the Umpire he will be fined for arguing,I think it is mandatory to have 11 fielder in the ground when the match is been played as if any one of them is absent it will be a no ball for each delivery with 10 fielders in the field.
But there is an incidence where whole team refused to take the field
The Pakistani cricket team toured England in the 2006 English cricket season, following England's winter tour of Pakistan where Pakistan had won a 3-Test series 2–0 and the ODI series 4–1. As a result of this series, Pakistan rose two spots to be ranked second in the world Test rankings before this series, and England dropped to third after the 12 July annual update of the ICC Test Championship. The series was marred by a ball-tampering controversy on the fourth day of the Fourth Test, which culminated in Pakistan refusing to take the field after the tea interval in protest to a contentious decision by the umpires to penalize the Pakistani team for what they believed were acts of ball-tampering. This led the umpires to award the game to England, who thus won the Test series 3–0. However, ICC in 2008 declared the result of the final test as a draw and England thus won the Test series 2-0. This was subsequently reversed by the ICC in February 2009 and the result restored as an England victory.
I have added extra information here just for more clarity.