Similarly, I want the ability to make a channel not ping the browser tab (no * next to it) but still display the channel name in bold if there are messages. I find it hard to ever remember to check some channels, but don't necessarily need to check them immediately (or even very quickly).

If you mute the channel and then configure slack to only show starred rooms and rooms with new messages, you can achieve this effect. The room will stay hidden unless someone says something, at which point it shows up in the list, unbolded. This lets you know to check on it at your convenience without triggering any of the notifications.