First, I want to say that this is shaping up to be an amazing game. I absolutely love all of what has transpired between the last alpha patch and the beta, and I am thoroughly impressed.
That being said, the Beta has generated a lot more crashes than any of my alpha play-throughs. In fact, I'm unable to finish my game on medium due to crashing.
Anyhow, here is my feedback thus far:
[crash] Shipyard Management - On a medium spiral galaxy with three enemies, starting at around turn 100, anything to do with shipyard management (especially adding sponsors) causes the game to crash. At times I cannot double-click a shipyard to enter the manage screen, other times I can. Sometimes I cannot click "manage" without it crashing. When I can get to the management screen, I can remove sponsors, but cannot add them without generating a crash. I did send crash reports via support ticket.
[crash] Large Galaxy - First attempt at large galaxy resulted in a crash when I double-clicked my homeworld to manage it. This happened twice, so I sent a crash report and tried a medium map. I have since been able to start a large galaxy.
[bug] Approval Numbers do not Match - In the picture below, I highlight that the numbers from the bottom left and the list on the right do not match. Also, the 'total' doesn't always match the math (in this case, +30 - 31 = 0).

[bug] Grid Showing Beyond Border - I really like the ability to toggle the grid on/on in ZOC/off. But when set to show only within ZOCs, it bleeds past the edge a little. It's a bit hard to see in the image below...

[bug] Range Still showing after Deselect - The range of ships is still occasionally showing in the FOW after the ship is deselected, as pictured below:

[bug] Black Holes showing through FOW at Hex Corners - It's probably a bit tough to see in the image below, but the black holes are bleeding through the FOW at the corners of the hexes.

[bug] ZOC on Edge of Map still showing through FOW - I know this is something that you are probably aware of, but I figured I'd also mention that enemy ZOC boundaries are still showing up through the fog of war when they are on the border of the map.
[bug] Missing Icon on Planet Screen - I'm not sure if it is the image that is missing, or if it is the particular bonus, or if it is just random... But the tile in the image below is missing an icon for the "precious metals" bonus.

[bug] Upside-down Path Numbers when Camera Fully Vertical - Whenever the camera is completely vertical (no tilt), the path numbers go upside-down.

[bug] Text Clipping - In a handful of places, text is being clipped:

[UPDATED] [bug] Selected Anchor not showing on Symmetry - After some more playing around with the designer, I think this function is working properly... but anchor selection is somewhat squirrely. But the new color-changing anchors are wonderful.
[annoying quirk] Moving Tooltip - In the (extremely awesome updated) tech tree, the tooltip moves vertically depending on where your mouse cursor is positioned. This isn't a big deal, but it can be annoying if you are reading and the tooltip doesn't just slide a little, but instead flips above or below your cursor's vertical position...

[suggestion] Tech Selection - In the tech tree, it would be nice to be able to double-click to select a tech (and thereby exit tech tree screen).
[suggestion] Planet Governor Production Slider - Add icons like in the main governor to indicate manufacturing versus military production.
[suggestion] Planet Income vs. Expenditures - Maybe I'm missing it, but I recommend adding a profit/loss indicator on the planet screen. This is present in the galaxy view tooltip, but it is not available in the planet management screen or planetary governor.
[suggestion] Increase Shipyard Sponsor List Scroll Speed - I know this isn't as important as hammering out the crashes related to adding sponsors, but the scroll speed could be increased (as it was in the planet build queue). Also, it would be nice to have the ability to add/remove more than one at a time, or to switch them from one shipyard to another.
[annoying quirk] Colonization and Survey Event Overlap -When a colonization event occurs during a turn, it is often interrupted by a survey event--which is fairly annoying.
[curiosity] Colonial Hospital - Not sure if this is supposed to happen, but when I upgrade a colonial hospital (one per planet) to a xeno medical center, I can once again build a colonial hospital (but not upgrade it due to the xeno medical center being one per planet, I assume).
[suggestion] Soil Enhancement - It seemed as though the soil enhancement improvement worked on about a third of the planets I colonized. I assume the rest didn't have sections of sufficient land mass to allow the improvement. In such scenarios, it is annoying having the improvement listed but unable to be used. I suggest either a ) make soil enhancement always work by forcing the hex with the most land to be usable (i.e. if none meet the requirement, the one that is closest is available for the improvement) or b ) make soil enhancement unavailable perhaps by graying it out on the available projects pane and adding a tooltip that says "no area on the planet meets the requirements for soil enhancement."
[suggestion] Shipyard Queue - I like the queue, but I am not sure I like that I cannot upgrade a ship mid-build. For example, in GCII it was possible to switch what I was building without losing all the previous production, which enabled me to complete research while building a ship, then add the newly-researched element to the ship by making an upgraded version. In other words, If I have 2 turns left on a colony ship M1 and can now build a colony ship M2, it would be nice to have the option to switch the M1 to M2 and just add the difference in production time.
[praise] New Visuals - I love all the new visual elements, including but not limited to the improved ship graphics, the planet tile icons, the galaxy-view icons (resources, ships, bases, etc.), and the grid overlay on the buildings in progress.
[praise] Improvement Building Streamlined - I love that I can now simply select an improvement from the "available improvements" list and double-click them into place one after the other.
[praise] Auto-upgrade - I love that auto-upgrade for improvements has been added, and I love that the toggle for this option is on a planet-by-planet basis.
[praise] New Events - I love the new events (though scorpions aren't insects
). One of my favorites thus far is the one where there is "a sudden increase in storm activity" which causes the "Leaking Seals." I love it not because it is good for the civ (only negative options) but because it forces you to choose which negative option. And I love it because "We can't afford that. And even if we could, we don't want to coddle these people. They'll never develop a tolerance for poison that way." LOL!
So much great stuff--keep it up SD!