Depth of cut tends to be one of the most important aspects to using a router that is often overlooked.
Generally speaking, you're better off taking very thin cuts when using a router. This allows for consistent feed speeds without bogging down the router bit speed.
If you're working with woods of varying density (ie. knots in the wood) then you'll want to size your cut depth based upon the worst case of what you expect to see.
The balancing act is taking lots of thin cuts takes more time and we often want to move on to the next portion of the project. So sometimes you cut a little deeper and take the risk of scorching, knowing that you'll have to sand that out later.
To address some of your questions:
Should you see better results using a routing table instead of locking the board and using the router freehand?
A router table can provide better support for the material so you'll have more consistent results. But some pieces won't fit on a router table and you need to go freehand instead.
Age of the bit isn't as much of a question as sharpness of the bit. Given that older bits tend to be more used and therefore less sharp, newer, sharper bits are the better way to go.
- Speed of the router (or power output), does it make a difference?
You need to adjust the speed of the router based upon the size of the cut you're making. Big bits need to be run at lower speeds. Smaller bits can be run at higher speeds. The primary concern is the friction generated between the bit surface and the wood.
Big bits run at high speeds generate more heat from friction which leads to a greater chance of scorching the wood.