Want an advice from a game developer :

I'm not a game developer but know this.

There is a long list of different formats and depending from one game engine to another they typically uses one of those different formats (e.g. TGA, TIF, PNG, DDS, etc.). Besides that there are also different compression algorithms used by the different engines so it's hard to say what is best because it just depends on the game engine used.

From a standpoint of a stock media developer you want to avoid having all these different formats up for download. You basically just want to provide a format that is able to contain all the necessary data and leave the conversions up to the game developer and his specific target platform he is developing on.

Most engines work with the sRGB color space, 8 bit for colors is ok and usually 16 bit for normal maps because you would want to get in as mush data as possible and leave the developer to decide how hard they wants to degrade/compress the normal data.

Depth maps for terrains are also typically in 16bit and can be TIF, PNG or TGA.

The PNG format is a good format to store all your pixel data in, note not all files need to be in 16bit.
TIF or TGA are also good options, TGA however can generate smaller files without losing original data, but you can also just store your files in a zip or rar file and then things also get a bit smaller.

Only a modder of older games here.
Most game Engines love to use a .TGA file format and many use .DDS formats.
I prefer to have a model that uses
one diffuse at least 1024x1024
one combination Environment/Specular map
and finally a normal map to finish things off. (normal map can be half size)

Most game engines Require something like a power ratio of 2 to work optimally .
I would recommend you stick to these ratios below should you plan to release work for games

1024x1024
2048x2048 <---- Unity Recommended
4096x4096 <---- UDK3

512x512 <----- low poly engines SAGE uses this size
128x128 <----- Super low poly engines 001 uses this size as does a few others like RPG maker