In English, I can say
The running dog eats.
This is a proper sentence and isn’t very weird or awkward wording. It uses “running” in the place an adjective could go.
In Spanish, is
El perro corriendo come.
or something similar proper grammar?
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Theoretically such a form exists, and it's called the participio activo. It was extremely common back in the day, and could even take object pronouns.
However in modern Spanish, it's no longer a proper verb form and is really just an adjectival ending. The ending is -ante/(i)ente. The problem that you'll get in using it is that it really only can refer to attributes (things you'd describe with ser) and not states (things you'd describe with estar).
Thus, it's perfectly appropriate to talk about la Bella Durmiente (Sleeping Beauty) or a caballero andante (lit. "walking knight"). But the adjective running needs to be reasonably considered as intrinsic to its being (such as the case would be in a flying squirrel which could be called an ardilla volante1, regardless whether it's in flight at any given moment, or a planta flotante that may or may not have grown large enough to float yet or may have sunk after dying).
In your case, I don't think that running is an intrinsic characteristic (besides which corriente has some other more common uses). Thus the simplest and most universal way to translate that idea is to simply add on a que [verbo].
El perro que corre come.2
Make sure to match tenses
El perro que corría comía
This is very common and also very natural sounding. If you really desire an adjective, you'll have to content yourself with not having a precise rule. Sometimes the ending might be -dor, other times -ado, or some entirely different word. It's only semipredictable. In some very rare cases, you may find -ando/iendo used (!), but these will almost always have been nominalized (turned into nouns).
1. Nowadays, the more common translation is ardilla voladora, but volante has some historical use in scientific literature.
2. If you feel that your statement may be confused for a gnomic verb (stating a generic truth, e.g. "the early bird gets the worm"), you may want to throw it into a progressive, but it's not strictly necessary.
El "presente progresivo" en español se forma con el verbo estar y con el gerundio del verbo que hace la acción.
Yo estoy hablando contigo
Tú estás leyendo mi post
El gerundio es una forma verbal con valor adverbial. No posee valor adjetivo ni sustantivo, como sí tiene en otras lenguas (como el inglés).
El gerundio debe cumplir tres condiciones para que su uso sea considerado correcto, según este planteamiento normativo:
Not the way you proposed, but....
(I have a hard time imagining a dog running and eating at the same time, so allow me to tweak your sentence a little.)
A sleeping dog does not guard the house well.
The closest we can come to this in Spanish is
Un perro, durmiendo, no protege bien la casa.
Note that if we backtranslate, we get
A dog, sleeping, doesn't guard the house well.
A more natural expression than the slightly silly "Un perro, durmiendo" approach:
Un perro dormido no protege bien la casa.
But the pattern does fit well with some other things, for example:
Mi foto de tu perro durmiendo en el sol salió súper bien. | My photo of your dog sleeping in the sun came out great.
Similar to this, I can imagine a caption of a photo, or a title of a painting:
I do want to be clear that this is not exactly equivalent to "sleeping dog," but rather
Mostly in addition to what @aparente001 wrote, I'd like to share a very good summary on the correct and incorrect usage of the gerund that I found on the Internet.
In connection with the adjectival use, it is worth noting that, similarly to their use in captions, gerunds can also be used with nouns to describe situations (always with dynamic or action verbs):
There is also the interrogative and exclamatory use:
And the use after direct objects, where the gerund functions as an object complement, mainly after verbs of perception:
Finally, though colloquial and not strictly correct (a relative clause would be preferred from a grammatical point of view), I'd also add this case of two nouns accompanied by dynamic gerunds in comparative or parallel structures:
The only gerunds that, according to the summary, are accepted are "hirviendo" (boiling) and "ardiendo" (burning), respectively equivalent to the adjectives "hirviente" and "ardiente".
Needless to say, all gerunds functioning as adjectives will be placed after the noun.
From A New Reference Grammar of Modern Spanish Chapter 24:
The Spanish gerund is a kind of adverb and it therefore should theoretically not modify nouns...the gerund is usually possible only when there is a verb in the main clause to which it can refer, e.g., me escribió pidiéndome que fuera a verla...But this rule is broken...constantly in spontaneous speech and informal writing.
Some of the examples offered in the book: - in captions to pictures:
'Dos cazas siendo preparados para el despegue'
on a sign: 'Hombres trabajando a 400m'
'...volvió en sí con el brazo sangrando'
'...con la luna ahí colgando para nosotros'
'Tenía mi edad y un hijo viviendo con su mamá'