1) Doubt anyone cares as long as you're getting your work done well, and you're responsive & available (as anyone working remotely on a computer all day should be).
2) Some companies require you to disclose any potential conflicts or other business interests. Many are not diligent about keeping that updated, but probably require you to disclose it if there are changes. Might set off some red flags if you ask, and otherwise could be a terminable offense if they find out (but how would they find out?)
Also, be aware of productivity tracking software out there, either installed directly on your work-provided PC or applications you're using over network.