Is it true that GL SE can't have its logo as long as it has 'beta' status?
(I know that there were several related questions before, but none of them are as specific as this question, and some of them are obviously obsolete)
Getting a custom design is one of the perks of site graduation. It used to be that getting the design happened at the same time as graduation, but Stack Exchange changed that policy some time ago: see Design-Independent Graduation is on for early September! and other questions about "design-independent graduation" on Stack Exchange Meta for more information.
So, yes, it's true that this site won't get a design until after graduation. It might be a long time after graduation too. As an example, Motor Vehicle Maintenance & Repair graduated in September 2016 and doesn't have a design yet.
We have some news there. There is some discussion on SE, and we are also nominated on such proposal: Let's break up with "Graduation" and remove a bunch of "Beta" labels.
In short: maybe stable and long live beta sites will have an upgrade (from beta), but not yet with full perks (custom design/logo).