Yes, they can all live together if the pups are females and the father is fixed. But not in that pet store cage. It's not big enough for one pig, much less four. Click on the Main tab at the top of this page for recommended cage sizes, and those are minimums. Bigger is always better.
You'll also have to keep the dad separated for four weeks after he's neutered to make sure all the sperm die off.
A good sexing aid is
https://www.cavyspirit.com/sexing.htm. Or you can post pictures here. Make sure they're well-lit close-ups so we can actually see what's going on.
If one of the pups is male, you can have him live with dad, and leave the female with the sow. If they're both male, you can put all three males together in a large cage, and maybe they'll stay together. It'll be easier because the pups are so young, but three males can be problematic when the younger ones hit puberty.
In that case, the sow would be alone, but you could find her a friend. It won't work to have multiple males in with females. A very few people I know of have been able to make it work, but only if one of the males is very laid-back and non-dominant. Most of the time it ends in total, miserable failure.