Did the pope ever commission Mercator to make a map?

No, it was not commissioned by the Pope, but patronised by Lord Wilhelm Duke of Juillers.

The 1569 world map in question has a legend in Latin which has been translated as

TO THE MOST ILLUSTRIOUS AND CLEMENT PRINCE AND LORD WILHELM DUKE OF JUILLERS, OF CLEVES AND OF MONT, COUNT OF THE MARCHES AND OF RAVENSBURG, LORD OF RAVENSTEIN, this work, commenced and ended under his favourable patronage, was dedicated by Gerhard Mercator.
_{My bolding}

and there is no mention of the Pope. Additionally there is

To the readers of this chart, greetings

In making this representation of the world we had three preoccupations.

[Summarised below as]

Firstly, to spread on a plane the surface of the sphere in such a way that the positions of places shall correspond on all sides with each other both in so far as true direction and distance are concerned and as concerns correct longitudes and latitudes.

The second object at which we aimed was to represent the positions and the dimensions of the lands, as well as the distances of places, as much in conformity with very truth as it is possible so to do.

The third of our aims was to show which are the parts of the universe which were known to the ancients and to what extent they knew them.

Again, there is no specific mention of the spread of Christianity (apart from their invasion of Antioch), but the notes mention known history such as the Tartars, the Mongols, and numerous historical and geographical facts unrelated to the spread of Christianity. There is an attribution of the 1493 papal Bull arbitrating on the division between Spanish and Portuguese spheres of influence.

On finally, on copyright:

By prerogative of His Imperial Majesty all whomsoever in the Empire or in His hereditary Kingdoms and provinces, are restrained for a period of 14 years from reproducing this work in any manner whatsoever or to introduce it therein should it have been reproduced elsewhere. The same restraint has been promulgated by an edict of His Royal Majesty for a period of 10 years in Belgium.

A few copies of the map exist, for example in museums, but if there is one in the Vatican, it must be a secret.

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A map showing the spread of Christianity would almost certainly have been constructed, at that time, in a subjective style with either Rome or Jerusalem at its centre, and the Christian influence shown to spread in all directions.

Mercator's new map was in 1569 the most significant advance in the history of cartography, inspiring map historian Nordenskiöld to write "The master of Rupelmonde stands unsurpassed in the history of cartography since the time of Ptolemy." It was an evolution of navigation maps and charts.

I can find no references to the Pope having commissioned Mercator to make a map. Pope Gregory XIII did commission maps in 1580 from his cosmographer Ignazio Danti and these were for a gallery.

Gerardus Mercator was commissioned to make a pair of small globes for the Holy Roman Emperor, but no mention that I can find of a commission for the Pope.

If Mercator had been commissioned by the Pope, I suggest there would be frequent references to it. His other maps are well documented, but there is a lack of any mention of Papal commissions.

Mercator escaped arrest by the Inquisition, but was later caught and imprisoned for some months. He also had leanings towards the Lutheran Church, so it seems unlikely.

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It is true that the Mercator projection distorts the area of countries, but that is not why it was developed. It became a favourite among navigators, not because it made them look important, but because a straight line can be plotted which represents sailing in a steady compass direction.

In modern times other projections have gained popularity, such as the Peters Projection which the presenter of the video shows, but she glosses over the fact that is has different distortions. It is most accurate at latitudes 45° north and south, which as it happens, favours Europe, where Peters lived.

Which map projection is 'better' is a matter of opinion, and depends on the purpose it is for. The truly accurate map can only be a globe, but there remains a problem that only a part of it can be correctly seen at one time.

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The linked video seems to be a bit of a "double act". One of the "audience" is supplying cues, agreement and laughter which reinforce the points. The presenter uses ridicule to deprecate the Mercator projection map, by incorrectly saying it shows the equator to be crossing Iowa in USA. Her conflicting assertions that it was made because:

• The Pope commissioned Mercator to make a map that shows the spread of Christianity.

• The countries in which there are predominantly white people are larger than in the southern part of the world.

are mistaken.