Ground Plane Tips for Beginners
A ground plane should be used to provide a nice, solid, low impedance path to ground. Some people, and almost all hobbyists (generalising), use it as a lazy way to hook up the ground connection as obviously this is used in a lot of places. In an ideal world, this means no other traces through it, and direct connection to the relevant pins. You may (will) have to break these rules in a real 2-layer board, but you can get much closer on a 4 layer board.

In something like an Arduino clone, you can happily use a ground plane and it is likely to work fine. Likewise, you can split the plane with signals, especially if you have no sensitive analog components, on projects like this. It's good practice not to, of course, but don't get too hung up at this stage of learning.

Which leads to autorouter. As a (weakly held) general rule, don't use it! While it may find paths for you, it is likely that thinking more about your component placements and general layout will be far more beneficial as a beginner. The real power of the autorouter is actually for advanced users who have to route out 1000s of pins from tiny ICs. Typically, you lay out as much you can (be bothered) by hand - this will usually make you place components logically, rather than just spattered all over the board. For a design the size you are describing, don't use it. Personally, I route sensitive signals first (local decoupling, small signal analog, fast digital etc), then power/ground, then hook the remaining airwires.

On a single board, you will not have a ground loop. Unless for some reason you have two different power supplies feeding the board. Ground loops tend to be more of a problem in small signal analog, less so in digital designs and it's easier to isolate with digital protocols.