I've purchased some gog.com games that run in dosbox. However, when I install them they run in an installation of dosbox that comes with the game. Can I just copy the games to my existing dosbox installation? I've set up a lot of preferences there that I'd rather use. If I can't do that, is there another way to install them there?

DOS games are nothing but loose collections of files. As long as you keep them together, you can move them anywhere, anyhow. (This is why they sometimes used brutal copy protection schemes.)

I for example, refuse the original DOSbox as obsolete (and my distribution of choice stopped serving original Dosbox months ago), so all you have to do is make sure the configurations are converted to a modern type and keep in mind that some of your graphical edits may be redundant in Staging since it's way better at looking like an actual display.

Dosbox Staging also lets you chainload configurations, which I imagine would be useful for you too.

In addition to what dnovraD said, if you intend to use Dosbox-Staging, make sure you read https://www.dosbox-staging.org/getting-started/introduction/ as well. Staging's wiki is at https://github.com/dosbox-staging/dosbox-staging/wiki.

Yes it's easily done. I install all my DOS games (one per subfolder) under one C:\Games\DOS\ folder, then start DOSBox with a shortcut link like:-

"C:\Program Files (x86)\DOSBox\DOSBox.exe" -conf "C:\Games\DOS\Albion\Albion.conf" -noconsole -exit

It's worth familiarising yourself with conf files, basically DOSBox settings file where all the settings, different custom keymaps (the key / controller rebindings you see when you press CTRL-F1), etc, can be custom tailored to each game. You can reuse the .conf files GOG provide (typically named something like dosbox_xxxx.conf (general settings) and dosbox_xxxx_single.conf (settings specific to each game) or use your own.

Basically all the CPU cycle, audio, etc, settings along with how to start the game are stored there. Each one usually "Mounts" the game folder then starts the game (usually just by running the .exe). Some more complex CD-ROM games that expect to see a CD drive need an extra IMGMOUNT line that will "mount" an .iso as a CD drive as drive D:. Others (eg, Crystal Caves) might show a menu, "Press 1 to start Episode 1, press 2 to start Episode 2", etc, pre-game. Once you get the hang of it it's fairly simple to use your own DOSBox install, use other DOSBox variants, eg, DOSBox Staging, or even put them onto a portable USB stick. Although you can copy GOG's own .conf files, it's worth learning how to do anyway to get the multitude of other DOS games GOG doesn't have (eg, Lemmings) running equally well.