There are already some excellent answers and suggestions given, but per request I'm posting my own.
First, the idea of an emotional support animal (ESA) seems to have legal meaning in the USA only. The Americans with Disabilities Act of 1990 prohibits discrimination against people with disabilities, both mental and physical. This is similar to the Civil ...
The English original is already quite clunky?
But as the concept catches on slowly in the German speaking world, a direct import for this kind of Assistenzhund* is usually explained as:
Emotional Support Animal / Dog
Ein Emotional Support Dog (ESD = Unterstützerhund bei emotionalen Unsicherheiten) unterstützt seinen Menschen im Alltag in belastenden ...
A common set phrase for "emotional support" in German is "emotionaler Beistand". Building on this, you could maybe use the phrase "Beistandshund" or "Beistandstier".
This is probably not a well-established word, but a quick search shows that it is at least occasionally used with the intended meaning. However, there are ...
even when it is defined in DWDS more as recipient of extended caressing. I would consider it as implied, that there is some emotional benefit on the other side as well. The noun is quite focused on a dog, however, and I see no more generic noun with the same flavour.
It is also implied, that the dog is smallish, so it can conveniently ...
Although you ask for a German version of "ESA" which seems to have a therapeutical context, I suggest
Seelenhund (soul dog).
This is a dog being in harmony with a person's soul and making feel human beings better. It is not an official term appearing in legal documents, but I think it covers the idea.
There is a clear distinction in German between
a) Es ist nichts
b) es ist ein Nichts
with a) meaning the same as the English "it's nothing", while b) would rather mean "void" (like in the great, big void between the stars == das große Nichts zwischen den Sternen) in the sense of the significant absence of anything.
If you want to ...
There is a construction that goes like this:
NP, X zu Y
where NP is some noun phrase (usually just a noun), Y in an infinitive, and X is a complement of the verb in Y (mostly an object or prepositional phrase). Examples:
Die Kunst, mit Fremden zu sprechen.
Die Erfahrung, in einem Restaurant zu arbeiten.
Die Fertigkeit, Motorräder zu reparieren.
The pronoun nichts means nothing, i.e. nicht etwas.
The noun "das Nichts" has a number of quite different meanings. The DWDS entry is enlightening. I'll attempt to come up with an English translation, but it's probably better to read the original. I'll leave it to you to compare to the English use.
a) (Philosophy) absolute non-existence, opposite ...