I got a TV with eARC/ARC support and an AppleTV 4K 2nd generation plugged via a HDMI switch, will it be able to output sound through HomePod?

Yes see: Use HDMI ARC or eARC with your Apple TV 4K - Apple Support

You plug the HDMI cord from your Apple TV directly into the arc port on the tv.

Apple TV needs to be assigned to the same room as HomePod and you should set the HomePod to default audio in ATV. settings and set up AIrPlay & Homekit.

Set up home theater audio with HomePod or HomePod mini and Apple TV 4K - Apple Support

In your tv settings for external inputs you can change that to ARC and use HomePods for all tv audio, not just Apple TV

Without a doubt, the ATV4K will be able to default output sound from ATV streaming apps to the HomePod(s) provided they're configured to be in the same room within the Home app.

The ATV4K should be able to output sound from the TV, and other HDMI devices plugged into the TV, if your HDMI switch is up to the job (HDMI v2.1 with support for eARC). It's possible that a lesser switch would mostly work, but choke on advanced audio streams like Dolby Atmos. TV settings for things like digital audio pass-through will likely be important.