Is this monitor good for gaming?

Only for casual games. The 60 Hz refresh rate and 6 ms response time are fine for slower titles but not for competitive shooters, where faster panels are better.

Does it have built-in speakers?

No. You will need external speakers or headphones connected to your computer.

Can I adjust the height of the screen?

No. The stand tilts only, so screen height is fixed unless you use a VESA mount or arm.

Is 1080p sharp enough at 23.8 inches?

Yes for office work and video. Text is crisp at normal viewing distance, though 1440p or 4K panels look sharper at this size.

What connections does it have?

It is a standard office display with video input for a computer; check the port list for the exact connectors before buying if you need HDMI or DisplayPort specifically.

Is it worth it over a cheaper 24-inch monitor?

If you want a plain, reliable 1080p screen from a business line, yes. If you need height adjustment, speakers or a USB hub, a rival with those features is the better fit.