Can I connect my arduino ground to an external battery ground? Are they the same when said battery is powering the arduino?
A battery's ground is floating. That means that it's "happy" to take on any absolute value by simply connecting it to another ground - it just stops floating. This is why it's easy to connect a battery-powered phone directly to a computer to get the photos off or whatever.

The only time you shouldn't connect two grounds is if neither are floating. A good example would be the grounds of two different houses, which are physically connected to the Earth (as in the planet). It'd be difficult to budge the whole Earth's ground level!

But shouldn't they be the same? Unfortunately not. You could measure a distinct voltage difference (sometimes in the hundreds of volts) between two house earths due to the differences in their makeup.

One small note: if you do connect two grounds, only connect them in one spot. If you have two other ground pins a small distance away from the connected ones, there may be a voltage difference between them: now the two are explicitly relative to each other, so neither can be considered floating relative to each other. The pair as a system - if neither are earthed at least - is still floating.

These so-called "ground loops' can induce current where none is desired.