1. It streams data, but does it also share codecs and other installed file support software?
No.

2. For instance, if file format support is installed on the media server computer, and a file is streamed to the TV, can that TV make use of the codecs installed on the media server computer?
No.

3. I've noticed something similar with the Xbox 360. It doesn't run MKV natively, but if a couple of plugins are installed on the computer, it will work through the Xbox 360 Media Center.
That's a MSFT thing. We are in the LG Forum where it's not the same system.

4. I've been reading about transcoding. This would explain the codec sharing, if this is in fact what is happening. A short explanation about transcoding, coming from my research is, that if the playback device (LG TV) doesn't recognise the file being streamed by the server (computer), the software (plex) transcodes the file and then streams it to the playback device in a format that it does support.

That's not codec sharing. It's transcoding. Don't confuse such a thing.

5. If this is true, is it also true that the hardware in the server (computer), influences the ability to transcode and stream files more efficiently?

STOP. RIGHT THERE. What is the word efficiently doing here. We have to get it to work first. Later we can address efficiency such as "Watts per minute of transcoding."

6. So in other words, if you stream files that would normally get the 'file is invalid' error, with this transcoding functionality, can this error be prevented by upgrading the servers (computer) hardware? Like put in a faster CPU?

No.
Bob