Copy the whole directory and sub-directories to your hard disk. Do not copy only the .MTS files, that won't work.
Do I need the entire tree (see http://en.wikipedia.org/wiki/AVCHD)?
Or do I only need CLIPINF and STREAM?
What's in CLIPINF? The one thing I care most is timestamp.
Also, if I cut the mts file (making it shorter), will CPI file needs to be updated?
Unless you like troubleshooting and running into walls, just copy everything. Do not try to limit it only to some subdirectories. Why would you? Are you strapped for disk space? If so, get some more disks.
What is CPI?