About godot, I know at least thrive which is a pretty interesting, resource heavy game (since it have many entities to handle) which works fine on all "my" computers (desktop, laptop, and my employer's one
which is pretty lame)
About engines, I would advice against Ogre3D, as I'm not aware about *any* FOSS game using it and having been finished. There might be some around, I'm just not aware of them. Also, it's only about rendering. And there, it suffers enough problems so that OpenMW (a game engine) which used it switched toward OpenSceneGraph, despite all the work this switch implied. IIRC, the major reason was that Ogre3D was unable to combine different models or something like that. I do not know the details.
Anyway, if you want to make a 1st person RGP, openMW is a game engine that you could use. It's actively developped. One problem there is, while they do work on a FOSS set of data (models, animations, textures, most notably) that they call "example suite" (go see their forums) and do intend to "de-hardcode" the engine, it's still in practice a morrowind engine reimplementation.
As for Quake's derivatives, if you go for that way, you'll need to learn QuakeC, which might or not be a problem. QuakeC is a non-standard C, I've been told. OTOH, depending on the version of quake engine you use, you can do very different things... by modifying the engine, as UFO:AI shows. If you want to use only standard languages, and still be with Quake-related tech, then you probably should take a look at deamon-engine. Which suffers a lack of documentation problem, probably as most FOSS engines that are used for an actual game. At least those engines are all able to do multi-player, and pretty moddable.
Another engine that can be mentioned is probable flare-rpg, if you want to create a hack'n slash, diablo-like game. It's only for single-player though.
Wesnoth is a game, but some people modded to death, so that they ended up with a different game, so I guess you can also consider this an engine. Targeted at turn-based games, and able to do multiplayer.
Now, for pure engines which let you a bigger freedom, I can name, without having any experience about them:
* panda 3D
* O3DE
* Torque
The only one of those I know a FOSS game using it being Torque, which shows that the engine can do neat drawings, if the user have the computer to go with that. Others I have no idea what they're able to.
Hope this helps.