Playing today's Beta 5, opt-in patch 2 as the Iridium. I've played a few times in the beta, but not made too many comments. Here's a list of issues I encountered today, as well as a few quality of life suggestions.
I know several of these have been noted before, but I wanted to chime in too. 

Bugs:
--Iridium start with 2 trade routes, but cannot build a freighter until the first trade tech is researched

--If you colonize two planets in the same turn with ships on auto-move (the ships arrive at both the planets before the colonize event appears), then you see the first colonize event pop-up, but not the second.

--Updating the planet specific Govern setting (Manuf/Wealth/Research) does not update completion time of improvements that are already in the queue (until you exit the screen).

--Shipyard: I ended up with "6/5" sponsors. I believe this was because the shipyard already had 5 sponsors, and then I colonized the "Mars" planet in the system which was really close to the shipyard: it then auto-sponsored the shipyard.

--Starbase requesting constructors: I thought the idea was that any specific ships added to the shipyard queue after a starbase has requested constructors would be added to the top of the queue. This did not happen for me. A possible confounding event: One (out of three) requested constructors had already been sent and arrived, however I "passed" and ejected the constructor to send it elsewhere.

--Combat: 5 ships versus a single starbase. Despite what the combat viewer showed (including destruction of some of the ships), at the end I had all 5 ships left, at perfect health. I may have to go reload the savegame to really get the details of this one, but I know that something was goofy.

--Rally Points: ships are not visible (hidden under the rally point icon), and also cannot be moved. I had to move the rally point to be able to give the fleet commands.

Minor Bugs:
--Lantern minor race's leader is named "DL Bradley"!

--the sound would get crackly on the Diplomatic screen, but I haven't had any issues on other screens or in general

Quality of Life
--Ala Endless Space, it would be nice to be able to exit a screen simply by right-clicking on non-interactive portions of the screen.

--Colonize event screen should list the planet name and it should be prominent. I have seen some suggestions of also including the planet position... perhaps with the minimap? How about a button that let's us delay the colonize event until we've looked at the map!

-Starbases
--Economic ring adds a bonus to Production (not to be confused with manufacturing). Where on the planet screen can I see that the planet has this bonus?

--Using the schafer button (or tabbing to the next event) to take care of an idle colony: would be nice if the map scrolled to the planet and gave you a second to see what it was, and where, before auto-opening the main planet screen. Often I am not sure where the planet was, which of course helps clue the player to what it needs.

-Tech Tree
--On the initial screen, double clicking on a tech will select it and close the screen. It would be nice if the same were true when looking at the full tech tree
--Double clicking on one of the 4 category icons (like Weapons, Colonizing, Economy, etc) should open the full tech tree
--As mentioned many times before, enable a toggle to go directly to the full tech tree if the player wants

--Planetary Governors: I know these are not in yet, but I have high hopes that you will be able to (at least) select the Governor from the "Colonies" management screen (accessed through the main Govern screen) and/or directly access the planet specific economy wheel from this list.