Learn German: Wir sind gerade am Zocken! - We are gaming right now!

This is the core of the colloquial progressive aspect. The construction is formed using a conjugated form of sein (here, sind) + am (which is a contraction of an dem) + a nominalized infinitive. Because the verb is being turned into a noun and follows a definite article, Zocken must be capitalized.

Is the am-Progressiv construction considered grammatically correct in all situations?

No, it is primarily used in spoken German, regional dialects, and informal writing like text messages. It is not considered standard for formal written German. In a formal context, you would just use the normal present tense to express this idea, often relying entirely on adverbs for the progressive feel.

Since am Zocken already means "are gaming", does adding gerade change the meaning?

It does not change the core meaning, but it acts as a very natural intensifier. While wir sind am Zocken already shows an ongoing action, adding gerade emphasizes that you are right in the middle of it at this exact second, much like saying "We are gaming right now."

What exactly does the verb zocken mean, and is it common?