If a managed network switch were to overheat, would you consider it no longer reliable?

What you should do is run more intensive monitoring on it than you usually would for a while. If you catch it acting up, you replace it. If not, you make a note of it in wherever you document such things and move on.

Most likely, the switch simply heated up beyond its temperature operating range (which is why you saw issues), but not enough to cause permanent damage (which is why it works now).

As noted by Tom, most modern hardware will shut itself down before it reaches a threshold where permanent damage is expected, so it's probably fine now that the fan's replaced - just keep a close eye on it to make sure, and make a note of it so no one's surprised when/if it dies a little earlier than the other switches, many years from now.

Potentially yes.

If it got hot enough, it's plausible that it could melt the solder on the chips, or even on the port connectors themselves. There's an interesting video on Youtube of someone deliberately overheating an Xbox in order to get the solder to melt enough to reflow the dry joints.

If your switch got that hot, it's definitely possible it could cause long-term damage, even though the fan has now been replaced. That said, most hardware will turn itself off when the internal temperature gets too high. It's entirely possible that this switch doesn't have that capability though.

See if you can get a full replacement under warranty from the manufacturer.