None of this is to say that you should never recodify the statutes. But, a basically aesthetic motivation that mostly matters to people who make up a tiny portion of the main users of the product (non-lawyers) that poses multiple threats to how statutes will be located using legal research tools, and how it will be interpreted once located, meanmeans that reorganizing statutes is not something to be done lightly.
As a result, recodificationre-codification is most often done only as part of an overall omnibus reform of an area of law on the merits where the interest in being able to track prior interpretive caselaw and determine the original meaning of a statute enacted long ago is at its lowest.
Barring the even more ridiculous case of Congress repealing everything and passing the exact same laws again, just under different Titles/Chapters/etc,
This is not so ridiculous.
Most states have done exactly that one or two times in their history (sometimes more for older states). Also, many newly admitted to the union states also do something similar. For example, the basic template for the statutes in Colorado was the statutes of the state of Illinois. Basically, the first time around, somebody started with an Illinois statute book, eliminated Illinois specific laws and laws that they didn't like, rearranged them and adopted them as the original statutes of the State of Colorado (it may have actually been version 2.0 after a stopgap set of statutes, I don't recall which).
Also, most states completely repeal and readopt all of their statutes in codified form on the recommendation of the codifier to a legislative committee, to reflect the acts passed in the previous session, every year. This doesn't involve overhauls really, but in principle is a complete repeal and reenactment.
While the federal government has never overhauled the entire U.S. Code in this manner, it has been done at the level of individual titles of the U.S. Code.
For example, the most recent major reorganization and overhaul of the structure of the Internal Revenue Code (Title 26) was in 1986. The copyright laws were overhauled in the late 1970s. Both of those accompanied major substantive changes in those areas of law.
Another major recodification of many parts of the U.S. Code took place following 9-11 in connection with the creation of the Department of Homeland Security, which reorganized both the bureaucracy and the associated U.S. Code language associated with dozens of federal agencies.
would it be possible for Congress to arbitrarily merge, combine, and delete Titles, and to rearrange the location of laws, definitions, etc? Are there laws/regulations governing this?
Yes. Congress can do so any time it wants by passing a bill (it would be the longest bill ever - dozens of volumes long) doing so, and getting the President to sign it. The legislative process would be exactly the same as any other bill.