Are my circuits properly grounded?
No, that's not right.  -- Unless UL says it is (i.e. if they approved the labeling and instructions for the lug to say that you can do that). But I really doubt that. 

Retrofit a ground bar.  They are readily available in the $7 range.  Your panel probably has holes pre-drilled and pre-tapped for an accessory ground bar; add a photo of your panel's labeling and someone (ThreePhaseEel :) might even be able to identify the particular Eaton part number that will fit right up. 

Then attach the wires 1 per hole, or whatever the ground bar's labeling/instructions tell you to do. 

About separate bars

In a perfect installation, this N-G bond is a heavy wire (or several, redundancy allowed) that, ahem, you can put a clamp meter around.  

In a best-practice installation, they may skip that, but still put neutrals and grounds on separate bars -- not least, this makes it easy to convert a main to a subpanel.

Worst-practice (but still legal) is to simply declare all bars neutral+ground bars, and simply land all neutrals+grounds onto the nearest convenient bar.  Obviously, that makes it a lot harder to put a clamp meter around the neutral-ground equipotential bond, or convert to a subpanel. 

So it looks like somebody started out doing the "best practice" version of this, with neutral on one bar and grounds on the other.... and somebody else came along and added a circuit with the latter doctrine. 

Since fully separated N-G has its usefulness, and it's 95% done already, I would maintain that in this panel. 

Those MWBCs

One other thing.  Judging by the red+black wires, it looks like you have some multi-wire branch circuits.  From the numbered stickers being out of sequence, it also looks like breakers have been moved around.  This combination is a recipe for disaster.  To fix it, follow each red wire back to its origin cable, identify the red+black pair from that cable, then follow them back to their respective breakers.  Make sure the breakers are adjacent (240V between them) then add a handle-tie ($3) or replace both breakers with a 2-pole ($9).