Yes, he will end up realizing he's part of the family and start to "own" his new environment. He has to. Why he wasn't barking much in the first weeks is probably for the same reason a new student stays "low profile" in a new school; shyness and not knowing the rules of the environment.
As the other answer mentioned, the dog has some rights to also express himself now and then, and indeed, without realizing it you may end up rewarding a behavior you want to prevent. You can't expect him to live a his own little world of silence. You should in fact be happy, the fact he expresses himself after a few weeks shows he probably feels comfortable and safe with you! So you certainly are doing things right.
I hope this will not sound patronizing in any way. Truth is I love dogs and have worked with them for years. So what follows is more to put things in context and maybe use the perspective of your dog.
Noticing too much when the dog barks exactly after so little time is already giving some clues about how much you maybe focus too much on this.
Be careful when stressing too much about this situation that is, quite frankly, very normal! Your attitude could in fact provoke the behaviors you want to avoid; what often happens is, after a while of focusing on every sound that provoke barking, the owner starts being stressed even before the dog has any reason to react and this will make the dog bark. You might start a loop of stress-bark-stress. A relaxed pet owner who goes with the flow has less chances of having a dog who constantly barks. That's where it will get if you try to control too much how and when the dog should bark.
Understanding why he does it is different but take these analysis in their contexts too. Maybe try to have the same tolerance as parents have with kids, for example; they play, make noise, make the walls tremble, then get tired and when they're bored or stressed, they might react more to some stimuli! Your dog is a dog; he also needs to feel like one. What you may find useless or annoying is to him his way to tell the world "hey look, it's my home, I care about these people!" If it's not an aggressive or persistent bark, it should be seen in this way; a proud dog doing what dogs like to do!
Yes, he's probably realizing he needs to take his place in the family, and by instinct dogs are protectors. They in fact evolved with humans for thousands of years within that kind of relationship and "usefulness."
The other thing that is important to point out is:
Your dog is 7 years old and has known already a few homes.
Your dog is like a 50 years old human, he has his personality very well developed, his preferences, experiences, habits and he should be respected in the same way you will respect a human of that maturity and experience. He may be a dog but you can't expect him to be as "malleable" as a teenager.
You will end up totally loving him for this, you'll see! In fact, if you give him the time and freedom to show you who he is, you will regret in a few years not having known him sooner... Older dogs have a rich personality and with time you will enjoy discovering how unique he is.
The fact that he had many homes and lived in a shelter is also something that might change a bit the way he will behave, at least until he feels confident you're "the ones." You need to have empathy for his situation first and understand that this is very scary and may cause a lot of anxiety for him. He might not know exactly how he should behave and it's confusing for a dog to change home, they are animals who have a pack mentality.
I suggest you fully devote your attention to making him feel like part of the family and 100% accepted as he is first. Give him a break for a few weeks for the barking and really, stop taking notes on when he's barking exactly. You need to stop caring so much. Instead, react as it happens IF it's not acceptable. You don't want to start making the dog feel anxious and insecure because he doesn't know what to expect. Be consistent, smile when the dog does a little innocent wooo-ooooof at a cat and relax. If the dog needs to be shown to not bark, make eye contact and act as a pack leader; dogs read human faces very well and it's your attitude that will get you respected from your dog but also predict a lot of his behaviors if he's a "normal" dog (which he seems to be.)
There's a huge difference between a few bored barks or greetings and persistent barking. I think if he was already inclined to be a persistent barker, you would know; it would have been already noticeable from the first days! Plus, he's not a puppy.
Try this and ask this stack again in a few weeks or months if the situation really changed negatively.
Source: Cesar Millan's techniques, in general!