tl;dr Version

• All gardens have the same range of effect regardless of size
• Large gardens are more effective at increasing beauty than small gardens
• Four small gardens have an identical effect to one large garden
• The relationship is linear so 4 large gardens have the same effect as 16 small gardens
• Gardens take effect on beauty immediately on construction/demolition
• Beauty increase is proportional to area covered by garden
• Larger gardens are more cost-effective to build than smaller ones for same effect

If I pack a 4x4 grid of small gardens into the area the size of one large garden, does that give me 16x as much beauty as an equivalent large garden?

No - it gives you the same increase for four times the cost.

Methods:

To test this I first found an area of roughly equal beauty:

...then trashed it:

I then chose the spot just in front of the armoury (quite arbitrarily though there was space for all the gardens here and a wide enough area to see the effect). This is it's normal beauty:

With one small garden, the effect is very small both in terms of difference made and area affected:

However with one big garden, the effect is magnified locally but has no further range (look at the tile with scrub-land on it immediately behind the garbage dump - it is the same colour in both images:

With four small gardens (roughly equal in size to one big garden), the effect is exactly the same as one large garden in both area and magnitude:

As expected, the area with four large gardens has the largest local effect though again the same range (looking once more at the scrubland behind the garbage dump):

Finally, a 4*4 grid of the small gardens has an identical effect to four large gardens in intesnsity and range:

It's worth remembering that gardens are fixed cost, therefore it is most cost effective to build large gardens.

A few apologies:

The camera keeps shifting about as it's really difficult to get it in the same place twice. The gardens aren't perfectly aligned to the same grid either as this was also a pain to achieve, though it doesn't appear to have had any effect. Also the game crashed several times when taking the screenshots, each time doing so resetting the visual settings so some of the textures might be a little different from screenshot to screenshot. Finally the game absolutely refused to take the last screenshot (garden overload!) so it's a bit more obscured than the others - sorry!

I was always under the impression that it does not matter, though I don't have a reliable source that directly states that.

The official and inofficial documentation does not mention the impact of its location. Since it does mention the location for various other buildings for which it is in fact important, I'd assume that the place really does not matter.

In my games I did not observe different behavior with different locations, except for the usual impact on getting people to work in there (which applies to any building, obviously).