EDIT: I eventually found the solution and I decided to put it here in case anyone else googles this thread looking for answers.
JediQuacker gave me a big hint (thanks dude), which I missed in my hasty reading
To get credit for "searching for allies", you have to go back to fleet and somewhere on the main deck there will be a turn in box, click that and you finally get credit.
To get credit for each flashpoint for the weekly 5, it was a little different than what JediQuacker but he was mostly right. All flashpoints gave you a "flashpoint done" popup at the end (the one where it shows rewards and you could add friends) but that does not give you credit to the Weekly 5 Veteran Flashpoint mission. You need to click accept on additional "Flashpoint Quest Completion" popup. Now some flashpoints would give that to you automatically immediately after the "flashpoint done" but *SOME* required you to go back to fleet, find the deck with bunches of holoterminals for quest turn ins. Click one, go through the role play stuff, and finally you get the Quest Completion popup, and then it would count to the Weekly 5.
EDIT: Original post below.
I have 2 issues:
1. I keep getting Hammerpoint Station over and over again with the "random" group finder. I've done 10-15 flashpoints and seen only 2 or 3 different ones and nearly all of them are Hammerpoint Station.
2. I don't get credit towards the weekly missions.
I'm level 73 at this point and have been slowly progressing through the original story line for sith inquisitor. I'm just now almost finished with Alderaan and haven't gotten back to Dromund Kaas to do Darth/Lady Zash's ritual (no spoilers please, but I'm pretty sure I know what's coming). Is it just that I haven't done enough of the full story yet to "unlock" the other flashpoints? Or is RNGesus deciding to f--- with me?
Somewhere on the imperial fleet I picked up missions for the "searching for allies (do a random flashpoint)" and "weekly veteran flashpoint (complete 5 for the week)" and I've done like 10+ veteran flashpoints and gotten no progress for either of these. FYI: Hammerpoint is specifically mentioned as eligible for the weekly 5.
I click the minimap icon for group finder
Verteran Flashpoints is selected and if I look at the filter all 23 of 23 possible ones are checked (and all of those are listed as eligible for the weekly do 5.
The group starts, I enter, we punch dudes in the face, I get the "Mission Complete" popup at the end and click Accept
Then I either click the Exit Area on above the minimap or if there is an obvious door I exit through that
And no credit to these weeklies. Am I doing it wrong?
I just now, dropped both, and picked them back up again from the fleet priority mission box, did a random mission and no credit.
I do notice that often when I leave I'm still in the group and need to right-click my portrait and quit the group from that.