In Ancient or Modern Greek, do verb infinitives have a gender? Do they have declensions?

[Pedroski]

I believe, verb infinitives are nouns in Greek. In Greek, nouns often take a definite article. In Greek, nouns have gender. In Greek, nouns have declensions.

Do Greek verb infinitives have gender? Do they have declensions? Are they used with an article?

In German for instance, I can take any verb infinitive, write it with a capital letter and use the neuter article "das". Mostly, German will just change the article to indicate the case, the declension.

das Können: the ability
das Schlafen: the sleep

Spanish use el + infinitive, which would make all infinitives masculine.

Thanks in advance for any tips or advice!

[spiros]

Yes, they do have a gender (in the case of articular infinitive), for example "το ευ ζην" (ευ ζην → good life, used both in Ancient and Modern Greek). Modern Greek though does not use the infinitive very often, mostly with ancient Greek set phrases.
Infinitives do not usually have declensions (see link below for more on this), but the verb they belong to, does, of course.

The articular infinitive corresponds to a cognate verbal noun (in singular number only). It is preceded by the neuter singular article (τό, τοῦ, τῷ) and has the character and function of both a noun and a verbal form
Infinitive_(Ancient_Greek)
Does the Greek Language Have Infinitive Verbs? - Helinika

The lack of an infinitive. In Greek, verbal complementation in contexts where English would use an infinitive is typically formed with the help of finite (subjunctive) verb forms (e.g. θέλω να πάω, [ˈθelo na ˈpao], literally 'I-want that I-go', i.e. 'I want to go').
Modern Greek grammar - Wikipedia

[Pedroski]

Thank you very much, very helpful!

A strange coincidence: a Chinese word for run is 跑 pǎo , a lot like πάω
跑 - Wiktionary

However, Chinese has neither noun declensions nor verb inflections of any kind.