играя

Вместо того чтобы идти в школу, он остался дома, играя в видеоигры.
Instead of going to school, he stayed home playing video games.

Does it mean the same as and played?
Вместо того чтобы идти в школу, он остался дома и играл в видеоигры.

Short answer yes.

Long answer I don't really like the sentence with играя. Играя implies a continues action simultaneous with the main verb, which is остался. This is where the problem lies, остался is a completed action, but играя is not. Maybe it's a little bit of nitpicking on my part, but I'd not use this sentence.

Quote:

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Вместо того чтобы идти в школу, он остался дома, играя в видеоигры.

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It's a mistake. The participle (играя) is using wrong here. How iCake said "Играя implies a continues action simultaneous with the main verb, which is остался. This is where the problem lies, остался is a completed action, but играя is not."​ Your variant is correct.