Good to hear that you are in the house and starting to get a routine!
I have a small terrier and, frankly, she doesn't go for long distances. In this heat, she does even less. She has short legs, and we call her the "cheetah" - she goes really fast for short distances, and then she collapses like a toddler. Frankly, inside exercise is fine for her. We walk her mostly for the pee/poop needs and some socialization with neighbors and neighborhood dogs. I suspect your little dog would do fine in the same way.
We never had a fenced yard, not for this terrier or the last one. In fact, because we have some woods around us, we don't leave her out alone at all due to the possibility of coyotes. And a neighbor's dog was out in their yard, and was attacked and killed by a neighboring dog who was roaming loose (black lab mix - so never assume that a "tame" breed can't go nuts).
I would think that the half hour walk is fine for now as are the short one after work, and a quick, utilitarian walk at bedtime. If the dog gets some outside time on the lead in the yard when the kids are playing, that's great, but it's not vital. If she's around them inside, she's getting socialization and attention, and that's fine. I'm assuming the dog was in the yard at the old house but not necessarily running laps around the yard. So while it seems like she was out a lot more, she wasn't necessarily doing the doggie Olympics. So, weather permitting, you take her out for longer spells, but when it's horrible, you just don't.
Our 17 pound terriers are/were both walked in the morning, mid-afternoon to dinnertime, and before bed. The prior dog was good at letting us know if she needed to go, and the current one is terrible at it so we have to stay ahead of her. In the bitter cold, believe me, the dog wears a sweater (she doesn't have a double coat of fur, only a single, so she definitely shivers), and she goes out only until she does her business, then back in. In the horrible snow, we've been known to carry her to where she reliably poops. My advice is that you take the dog to the same place on every walk, and let her know that's her pee/poop spot. Take a couple of training treats in your pocket, and reward her when she goes in that spot. Choose a place you can get to in the snow - so for us, a back corner of the yard isn't it because we couldn't reliably get there in 2 feet of snow. So we use the street, but we're not on a super busy road. A couple of our neighbors just shovel a path to a back corner near the woods though - you could consider that. So work with the training treats for now, plus pats on the head, when she does what she's supposed to and where she's supposed to. She'll learn to "hold it" once she figures out that the house isn't her potty area.
For exercise, I suggest you get a soft doggie frisbee and some stuffed toys (if the dog isn't a chewer) or get some mini tennis balls from the pet store. Let the kids take turns exercising the dog inside - mine runs from family room to kitchen, chasing her toys. She goes for 5 minutes, then quits, but does it about 3 times a day. There's nothing magic about outside time other than the biological functions. She needs something to do during the day if you're all gone, but frankly, our dogs often used that time to nap a lot so they can be raring to go when people come home. There are some activity toys that keep a dog challenged mentally if alone a lot, but watch the endless treats lest she put on weight. It's really okay if they are alone for chunks of time, as long as you make good use of the times when you are together.