I think you should use sentence formed with exister

for your first example Is that a thing? could be translated as est-ce qu'une telle machine existe?

your second sentence could be translated as pourquoi les combinés machine à laver - sèche linge n'existent pas aux USA?

Your last sentence could be translated as J'ai essayer de leur en acheter une en téléphonant au lieu de passer par leur site internet, mais apparemment, c'est impossible

To me, there is no idiomatic way to translate to be a thing

"To be a thing" n’a pas de traduction exacte en français, en fonction du contexte on doit adapter la phrase:

I wish there were a coffee machine that made my coffee automatically at my normal time every day. Is that a thing?

J’aimerais une machine à café qui s’allume à mon réveil tout les jours. Est-ce que ça existe?

Why are combined washer-dryers not a thing in the US?

Pourquoi est-ce que les lave-linge/sèche-linge ne sont pas vendus aux US?

I tried to just buy one from them over the phone instead of going to the website, but apparently that’s not a thing.

J’ai essayé d’en acheter un par téléphone au lieu d’autiliser leur site mais apparemment ce n’est pas possible.

One possible translation would be the use of "se faire" but it does not always apply and often you would use a specific verb.

I wish there were a coffee machine that made my coffee automatically
at my normal time every day. Is that a thing?

J’aimerais une machine à café qui me fasse mon café à mon réveil tous les
jours. Est-ce que ça se fait?

But you should say "Est-ce que ça existe?".

Why are combined washer-dryers not a thing in the US?

Pourquoi est-ce que les machines à laver sèche linge ne se font pas aux US?

But you should say "Pourquoi est-ce que les machines à laver sèche linge n’existent pas aux US?".

I tried to just buy one from them over the phone instead of going to
the website, but apparently that’s not a thing.

J’ai essayé de leur en acheter un au téléphone au lieu de passer par
leur site internet mais apparemment, ça ne se fait pas.

But you should say "[…] mais apparemment c’est impossible.".

"Se faire" might convey a subtlety you may not want. The best example is the last sentence. "Ca ne se fait pas" may be understood as "It is rude to do so".

So I guess as @Flying_whale said, "there is no idiomatic way to translate to be a thing".

Taking into account a quick googling:

https://en.wiktionary.org/wiki/be_a_thing

(colloquial) to exist, or to be available, widespread, possible, or a
common practice

https://www.macmillandictionary.com/dictionary/british/be-a-thing

to be an idea, product, or activity that is generally known or
recognized

and the discussion here

https://english.stackexchange.com/questions/144171/the-idiom-be-a-thing

I would translate it by exister, être répandu, connu, reconnu, etc. depending on the context.

The other replies have already provided translations to your examples. As further examples:

Apparently, eco-friendly underwear is a thing now.

Apparemment, les sous-vêtements écologiques sont répandus maintenant.

It is Valentine’s Eve. Is that even a thing?

C’est la veille de la Saint-Valentin. Est-ce que ça se fête, au-moins, ici ?

Funilly enough Google uses…être une chose 🙂 !