Tenchi muyo is one of my all time faves. I have the complete set. So I figured I would give you all I know about the series. Some you might already know but I hope I can fill in any blanks.
It starts with tenchi muyo the origanal series. Note that unlike most other anime wich are based on manga it was instead a light noval. This gave creators Masaki Kajishima and Hiroki Hayashi more freedom to create there work but less money to back it up. So as we know the original series was cut short.
A couple years pass and Tenchi became pretty famous in Japan. Due to this they decided to add to the series creating Tenchi universe and then the comedy Tenchi in tokyo. During this time four Tenchi movies were spawned:
Tenchi in love
Tenchi for ever
Tenchi Muyo: Daughter of darkness
Tenchi Muyo in love 2 eternal memory
(Note eternal memory is extremly rare and mostly unkown
outside of Japan)
There were then several fan mini spin offs made, Pretty sammy, mihoshi's story and carnival day. (Note: Carnival day is actualy an extra piece to the origanal Tenchi Muyo series.)
Of course this was not enough. Finely they created The no need for Tenchi manga. As you stated earlier there was some things that seem a little light. Lets say this, if you get the manga No need for Tenchi read it. It is actually the filler and connection between Tenchi Muyo and Tenchi Muyo Ryo Oki.
Of course after that Tenchi Muyo Ryo Oki was made to finish off the Origanal Tenchi Muyo series. After which they followed that by GXP wich happens to take place in the same continuity as the origanal Tenchi muyo about five years after Tenchi Muyo Ryo Oki.
Also some ware in the middle of all this mess. Was Dual! Parallel Trouble Adventure. Even though the story line is diffrent, it was a project ment to connect to the origanal Tenchi Muyo story line but due to money the project was cut and then they ended the series early. Easter egg, goodness, Shinbu from dual is actualy the unit Kasuki ends up finding at the end of GXP. There was suppose to be more of a connection but that all they could get.
And finaly the last series up to date of the series is another pretty sami spin off called Sasami pretty girls club. I can imagine you don't want that one, I didn't either.
Also note the series Tenchi Muyo! OVA series, Hasegawa canon created by Hiroshi Negishi. Despite this and even though it used a few of the same characters such as Kiyone. It wasn't counted as apart of any tenchi continuity by Masaki Kajishima. Masaki evidently won. Aint directorial rivalries fun.
So here it is. As far as I know it, the whole thing. This info I got from a combo of what I new filled with what I found on wiki.
As for Tenchi the series earns my official seal of Badassitude.
As for your Bro I suggest Dual! El Hazard, Gurran Laggan, R.O.D., Arouni Kenshin, and full metel panic. Each one of these comes with an excellent manga that I think fits what you discribed. Enjoy.