TL;DR: In the castle barracks, and they won't have families.
Well, twelve people really isn't all that many. If they are to be continuously responsible for the safety of the King and his family (let's assume a wife and two children), they'd be just about able to have one bodyguard assigned to each of them continuously if they worked three shifts.
Note that this doesn't leave anyone available to patrol the grounds, guard the gates, etc; you really just have four guards on duty, four on alert (maintaining their gear, training, or whatever) and four sleeping. This has the following consequences:
- The King's Guard are not the only ones responsible for the King's safety; they have to have a regiment or so of House Guards who patrol the corridors, guard the gates and are available to be called on for support (e.g. providing further escorts while travelling). The King's Guards are probably able to issue commands to the men and officers of the House Guards at need.
- Even though the guardsmen may be allowed to marry, they likely will not do so. This is quite literally a full time job where they are always on duty or on call, and can't spare the time for raising a family.
- This is also basically "grunt work", so you will only have young guys serving this way.
This paints the following (likely) picture of how the King's Guard works:
The King's Guard are a group of twelve armsmen who are directly responsible for the bodily safety of the King and his immediate family. They are chosen from young men of good blood (likely second sons of various nobles who do not stand to inherit and seek this as a way of social advancement) at about the age of 15 based on some audition or something, train for five years and then may be appointed as a Guardsman.
Upon appointment, they serve for a period of twelve years after which they retire (possibly marry), are awarded a small fief for their service and for the rest of their lives may be called upon to serve as the King's trusted captains whenever the King needs to raise an army. This gives you a nice cycle where you exchange one of the guards each year and you have both older, more experienced and younger, more vigorous members in there, and getting married at 32 is not terribly late; in earlier times, it was common to put marriage off until one had accumulated sufficient property and standing to support a family, and men tended to marry relatively younger women than today.
A member of the King's Guard is simultaneously commisioned an officer in the House Guards (probably above company officers but below the commander-in-chief of that regiment), which gives them standing and the authority to call on the guardsmen at need.
They live in the palace, probably in their own barracks (a bit fancier than the common sort - they'll even have actual beds and such). They may have servants and squires to wait on them and help with the maintenance of their equipment, but since they work shifts literally 24/7 they can't move away from whoever they're supposed to be protecting pretty much ever.
The wolves will be kept either in a kennel if you expect to be needing them momentarily, or in an enclosure near the royal hunting grounds, where they may socialise and get exercise.