Does control of the galaxy mean the Alliance is running the show on other planets from behind the scenes the way they do on Earth, or are they more out in the open about their domination?

Historically, the Anunnaki have proceeded with their evil designs on other worlds through trickery more so than brute force, particularly when there was any kind of ability on the part of a would-be victim civilization to defend itself. One of the consequences of spiritual disconnection and a highly distorted elevation of their own importance has been to create a race of cowards, because courage is an attribute of character and a divine one at that. So they will rarely take risks but proceed by stealth and deception to disarm their opponents, to infiltrate and influence indirectly by technological means, but also through psychic attacks and intrusion to manipulate their victims, subliminally if possible, to soften them up, make them vulnerable, and to disrupt their civilization and throw it into disarray as a softening-up exercise. This will enable them to come in and establish a presence surreptitiously and gain a widening circle of influence and power until they eventually have an entire world in their control. Their ability to manipulate the psyche of other sentient beings gives them a way to have surrogates in positions of authority who will be loyal to them and not to their native culture. This infiltration is how they have gained power all along and enabled them to spread their poison and make other worlds fall in line with minimal risk to their own wellbeing and survival. The other members of the Alliance use variations on this theme with the Reptilians being more willing to engage in battle directly, being more savage in nature, as this appeals to them greatly. So they enjoy taking risks but only up to a point; they, too, are not eager to fight out in the open, having learned many ways to do it indirectly. Their ability as shapeshifters to become an imposter with the exact appearance, knowledge, and behavior of a member of another race of beings creates a perfect cover for them to infiltrate and begin to manipulate the powers that be in a world they wish to dominate, and this can enable a political takeover to pave the way for a more thorough imprisonment of the population or set it up for annihilation, as the case might be. So the extraterrestrials are old hands at this, all three races, and of course the Arcturians have their robotic alien Greys to be the worker bees. They can be called on to launch an assault if that is of strategic value, as they are expendable, and this makes them a formidable foe in their own right. This is what makes the Extraterrestrial Alliance so dangerous because the advanced capabilities of each race outmatch what humans can mount in the way of defenses, let alone an effective offense to oppose them—that is why you need divine help to counter their subjugation of you and work towards having them make a decision to withdraw from your world—that is the divine plan and is a good one because you will lose in any kind of outright combat against them.