I know failed apprentices are a special case in that they can help in a lab even though they no longer have the gift. My question is whether they are able to build labs. Is a functioning gift (and magic theory) required to build a lab? Can someone with a broken gift build one? For that matter can someone without a gift that has Magic Theory build / upgrade a lab?
This whole "special case" thing has shown up before. There are actually several cases where someone/something without the Gift/Gifted can assist in the lab. There is actually no general rule listed. Rather, there is just a list of cases in which lab assistance is allowed.
As for working on a lab, for working on and improving a lab there is generally only a Magic Theory requirement. There is no Gift requirement. However, I believe there may be more stringent requirements on improving a lab's Refinement.
ArM5 does require only a Magic Theory of 3 or higher to set up a laboratory, not the Gift. As ArM5 is handling only the case of setting up a simple standard lab, this makes a lot of sense to me.
Upgrading a lab (by Covenants rules) is subtly different, since (by p.118 box) taking over a lab with Refinement > 0 previously used by someone else usually (*see below) requires a season of work, may reduce its Refinement and runs the risk of acquiring certain Flaws.
If somebody with high Magic Theory upgrades a personalized laboratory for another, he first needs to take it over (applying the rules from p.118 box, with the resulting consequences), then do his thing, and then the lab owner needs to take over her modified lab again (applying the rules from p.118 box once more). If somebody upgrades a standard lab to a personalized one for another, at least the owner needs to retake her lab afterwards (by p.118 box).
So usually a magus will not commission an upgrade of his personal lab by somebody else.
At higher Refinement levels this is exacerbated by the requirement (p.110), that whoever increases a lab's Refinement, "as a rule of thumb" "must have worked in the lab for a number of years equal to the existing Refinement score". This leaves some wiggling space to let one's long time lab assistant upgrade one's lab. But a stranger would need to work years to get to know it. And what could an un-Gifted person do in the first place to learn how an existing lab works?
(*) There is an exception specified in p.118 box, relating to labs "designed for transient use (as the spare labs in Durenmar are for example)", which do not require take-over procedures. These labs are likely set up and refined following well-known standard procedures, yielding results known and expected by all people with sufficient Magic Theory. Hence an un-Gifted person with Magic Theory 4+ might also set up such a lab at Refinement 1.
Anyway, this already shows how un-Gifted people with Magic Theory can only do the boring routine work around setting up and upgrading standard labs of Refinement 0 or 1.
Well, first, having worked setting up equipment for professional scientists labs that I was not working in, this is totally unrealistic.
More importantly, this is not required by the rules. The familiarity part is only required for improving Refinement. That condition does not exist for adding Virtues.
I don't see a magus' lab bearing more than a very passing resemblance to a modern laboratory.
The fact that lab refinement requires seasons of getting-familiar I have no problem with. I envision magus' labs as having their own personality and resonances, and not behaving the same way for someone who hasn't 'got to know them.' This flies totally in the face of modern scientific cause-and-effect thinking, but to me that's what makes it magical.
I don't have a problem with virtue addition being done by someone else, as long as the magus is on hand to point out where things should go and to make minor adjustments. I see this as a very different task than refining a lab; the former is knocking a hole in a wall for a window or constructing a kennel for your three-headed dog. The latter is spending time feng shui'ing your lab up to your personal tastes, requirements, sigil, etc.
Once the lab has the necessary space, the requirements for adding Virtues vary enormously with the specific Virtues to be added. Defenses or a Grand Entrance just need a stonemason - and his Magic Theory doesn't matter. A Greater Horde will typically require an adventure to find, and some diplomacy to move into the lab. A legendary craftsman must be found to to make Flawless Tools. And adding a Regio usually requires a team for a Hermetic Architecture project, and lots of vis.
How to add Virtues to a lab is hence not important, when discussing, what an un-Gifted Magic Theory specialist can do for upgrading labs in the context of the Covenants lab rules. It is rather very distracting. And so I omitted that topic.
What are the two things you do with a lab to improve it, leaving out move to/make a bigger lab. One is to improve the Refinement. The other is to add Virtues. And which of those is more common? I would argue Virtues because they can be allowed by Size, Flaws, and Refinement, so you can have more Virtues than Refinement. So why would we leave off Virtues when discussing improvements?
You seem to be confusing building something with installing it properly. Either that or you're arguing house rules here. There is a very specific requirement to add those Virtues. The "person" (doesn't specify magus, as in other spots), must have a Magic Theory at least 3 greater than the Refinement. For example, you can hire any random person to build a statue, but positioning it just right so that it works magically with the rest of the laboratory takes understanding magic (having a high enough score in Magic Theory).
Now, Free Virtues are a separate issue, and those we could ignore if we would like.
That "general rule" from p.113 describes a necessary, not a sufficient requirement to add most Virtues to a lab. For that reason I didn't discuss it above, when answering the OP:
You are however right, that un-Gifted people with sufficient Magic Theory can also be useful for adding Virtues to a lab with the necessary space available.