No GTFS support at the moment, but some of the map layers such as the Public transport overlay (looks like this) show public transport lines & stops.

Well modRana currently uses tiled maps for the map layers and you can tell it to download an are either around where you are on around middle of whats currently visible on the map or around a route. You can set are radius and number of upper and lower zoom zoom levels to download. Just note that it might take some time & not all tile providers like batch tile downloading. Also the tiles can consume significant amount of storage if high level of detail is being downloaded.

And if you plan to download a lot of map tiles, make sure to use the "sqlite" tile storage backend - unlike the "files" backend the tiles can't be shared with other applications like Mappero, but the tiles consume much less space (the "sqlite" backend eliminates the wasted space issues for many small files stored on FAT32).

In the future when the planed vector map support is working, I would like to make it possible to download (& update) vector map data pack directly from modRana (eq. Berlin, Australia, Europe, etc.) but that is still a bit in the future.
Yeah, as long as you have Internet connectivity though. And not only address searches but also Wikipedia and local amenity searches (find me nearest pizza, pub, atm, parking, etc.).

BTW, modRana is open source and mostly written in Python - and I'm pretty much the only one working on it at the moment and I can only do so much. So if you consider a feature important but it is missing, the best way to get in into modRana quickly is to send me a patch.