Sounds to me that what you're actually after is Teracopy, I know it does CRC checking, but I think it may do some of the other things your were asking for.
It can act as a direct replacement for the Windows Copy window if you want it to.
As it seems that you've experienced the limitations of TeraCopy I've found a few other alternatives:
Copywhiz is payware, but has some auto-renaming features that look like what you're asking for. Most of it's other features seem to match up with what you're wanting too, such as copying only new or modifed files and log file creation.
TotalCopy seems to be a rather simplistic replacement, mainly catering for poor connections with it's auto-pause and restart feature.
KillCopy appears to have a slightly better configuration set, but looks like it lacks the auto-renaming feature you want. I looks to be able to log the copy procedure.
SuperCopier looks a bit simple, but may have some of the features you require... at least the changelog reads "File collision handling improvement: it is now possible to rename and to overwrite only different files"
Other than that I'd recommend getting a copy of SyncBack Freeware Edition as it may have all of the features you require and can be set up to schedule the copies so that you know the files are always in sync.