Action, to an extent, is due to the height of the bridge. Which does look like it could be lower. It's way higher than the bridges on my guitars, but I do like as low an action as possible. The strings look like maybe .011 - 054 ish, so again, there's not a lot in your favour for learning to play barre chords. They are heavy-ish and will present more tension making them difficult to hold down. G string problem may be because it's wound, and will be tight compared with a plain G, although wound is commonplace on acoustics.
If you don't want to change to lighter strings, tune down a tone. It will make the strings easier to barre. but the bridge needs lowering. If it's a fixed bridge with no adjustment, it's a job that can be done, but not then undone! Go too low, and there will be problems.