How accurate/realistic is the conception of 'the kindly slave owner'?

A few preliminary points:

• 'New York' was of course 'New Amsterdam' for the larger part of the mid 17th century (until 1664). This is important, as the status and nature of slavery (and of free blacks) changed in the years following the British takeover.
• From the 1640s, Dutch West India Company had a system of 'half-freedom' which could eventually lead to full freedom.
• 'Paternalistic' seems a more appropriate word than 'kindly' when talking about slave owners. The kindest thing a slave owner could do would be to free his/her slaves, in which case (of course) he/she would no longer be a slave owner.
• For context, in 1664, New Amsterdam had a population of around 1,500, of whom 300 were slaves. About one in eight citizens were slave owners.

Domestic Servants

Female domestic slaves were common during the Dutch colonial period:

Slave women, usually no more than one per household, aided white women (free and indentured) with cooking, cleaning, and child care.

Source: Leslie M. Harris, 'In the Shadow of Slavery: African Americans in New York City, 1626-1863' (University of Chicago Press, 2003)

Male slaves performed many kinds of tasks, including skilled ones, but it seems that most were used for physically demanding tasks. This does not preclude, of course, male slaves working as domestics. They almost certainly did. (see also this article)

Whipping

This did happen, but there is too little information available to say how common it was. Administering punishment was considered a very low status occupation so

In New Amsterdam, a slave named Pieter administered punishments including whipping, maiming, and execution.

Source: Harris

There was a case of a public whipping of a child for theft in 1661:

Her punishment was so harsh that one scholar has noted, “she seems to have been the only child the court ordered beaten in this way”

The scholar in question would appear to be Susanah Shaw Romney in “Intimate Networks and Children’s Survival in New Netherland in the Seventeenth Century,” Early American Studies: An Interdisciplinary Journal 7, no. 2 (Fall 2009)

As the colony had few laws regarding slaves, there may not have been a law concerning the (private) administration of punishments such as whipping by slave owners.

Slaves earning money for their labour

On your third point, slaves could receive wages but the evidence I've found relates to the Dutch West India Company (WIC) rather than to individual owners:

WIC remitted wages to at least five slaves who, in 1635, petitioned the metropolitan authorities in Holland to order the WIC to pay them for their labor

Source: Thelma Wills Foote, 'Black and White Manhattan: The History of Racial Formation in Colonial New York City' (Oxford University Press, 2004)

Children of slave and slave owner playing together

On your fourth point, given that

the northern urban slaves of colonial New York City generally lived in their masters’ houses and in this respect led more confined, closely monitored lives than their counterparts in the rural South,

it is not unlikely that black and white kids did play together. Also, racism was much less evident during the Dutch period:

Blacks and whites had coequal standing in the colonial courts, and free blacks were allowed to own property (Jews, however, were not). They intermarried freely with whites and in some cases owned white indentured servants.

Schools were also integrated:

As early as 1636, Dutch officials in New Amsterdam had educated black and white children in integrated schools.

Unfortunately, the source doesn't say if these black children were those of free parents, but it is evident that race was not a barrier to these children mingling.

Also, the Dutch Reform Church (which was strongly opposed to slavery) married and baptized Africans until 1655. Although not evidence of slave owner and slave kids playing together, this does (with other evidence in this post) indicate that the colonist at the time held a wide range of views; this makes generalizing difficult.

Freeing of Slaves

On your last point, freeing of slaves was (by the 2nd half of the 17th century at least) uncommon but not unheard of:

Few New York City slaveowners rewarded their slaves with the gift of freedom. In a rare act of voluntary manumission, the widow Christiana Cappoens stipulated in her will that after her death Isabell, her female slave, should be freed from bondage and receive manumission papers documenting her free status. Cappoens also left Isabell several items of modest value: one small gold hoop ring, one iron pot, one kettle, a bed, and pillows.

Christina Cappoens died in 1693, admittedly a little late for your time frame. Earlier, in 1626, the Dutch West India Company (WIC) granted 'half freedom' to 11 male slaves and their wives:

A written agreement stated that the black petitioners were “free and at liberty on the same footing as other free people here in New Netherland.” But according to the “half-freedom” bargain, the WIC manumitted the 11 enslaved male petitioners and their wives on the condition that they were obligated to labor for the WIC when called on to serve and that their offspring were the WIC’s property.

Later, the DWIC granted the same status to more than 20 other slaves, but these measures were in effect forced upon the company by the Indian wars and the desperate shortage of soldiers (many slaves had previously served in the military).

Also,

On September 4, 1664, on the eve of the English conquest, Petrus Stuyvesant granted the petition of eight men “praying to be manumitted and made entirely free.”

Petrus Stuyvesant was Director General of the New Netherland Colony (which included New Amsterdam, the capital). Some slaves of individual owners also gained their freedom:

Besides some WIC slaves, a few slaves of individual slaveowners managed to attain freedom. For example, in 1649, Manuel de Spanje purchased his freedom from Phillip Jansz Ringo for 300 guilders, to be paid over three years, and in 1654, Captain Pieter Jacobsz and Jan de Graue manumitted Bastiaen d’Angola...freed as a reward for faithful service....

However,

slaves found it more difficult to obtain freedom from individual owners than from the company

Finally, it is worth noting that

Slaves, white and black indentured servants, and free black and white workers in the seventeenth century held more rights and experiences in common in New Amsterdam, and indeed in North America, than would be true in the eighteenth and nineteenth centuries.

Source: Harris