Alex if you imagine the old slum tenements in the UK, all now demolished, here the buildings were arranged around a paved/concrete courtyard. That's a "gård", and the significant thing I think is that it was an enclosed space. This kind of construction is still common in Denmark (and Germany) but not here in the UK.
Yes, "a garden/the garden" is "en have/haven". It's easy to confuse this with the verb "at have" - "to have" so watch out for this. Also, it is similar to the word for sea - "et hav/havet"; note the different genders. The difference between sea and garden is easily heard when speaking, though:
A yard in the U.S. is usually a space on a property behind or in front of a building. If plants are cultivated in it, it rises to the status of a garden. There are internal courtyards surrounding by buildings that are only in concrete with drains for rainwater. There are junkyards where old cars and scrap iron are sold. A yard in front of a residence may just be green grass (a lawn). A barnyard could have chickens or goats wandering about ...
The differences in the physical characteristics of "yard" in the UK and in the US have often caught me out. The online Cambridge Dictionary gives for YARD: https://dictionary.cambridge.org/dictionary/english/yard
unit of measurement
work area (lumber yard, shipyard)
land next to a building "that usually has a hard surface and that is used for a special purpose" (prison yard)
under BACKYARD, in the UK a small space surrounded by a walls at the back of a house usually with a hard surface<pre>
US: a space at the back of a house, usually surrounded by a fence and covered with grass</pre>
And Americans often call a yard anything that has plants growing in it, whether back or front of the house.