Say I have a shared hosting account, and then I deleted/moved/edited some of my files using ftp client, will the hosting provider be able to trace it?
"It depends". That depends on the software they use and how it's configured.
In short... yes. They can log everything you do.
The more long answer is... it largely depends on the kind of "shared hosting" you're referring to. If you only have FTP access and no ssh access (not counting SFTP) then a simple flip through their ftp logs will tell who did what from what IP when. Most big-providers do log this stuff quite heavily due to the legal implications that may come back to them. (i.e. if some law-enforcement agency traces a child-molester's content back to a provider... the provider must be able to provide data on the customer who posted it there & when) If you reserve a "VPS" or virtual server instance, and your virtual instance is running a FTP server within it, then the logs are yours to store or not-store. That still wouldn't prevent the provider from logging ftp traffic... as FTP is done in clear-text.
Now... the other side of this coin is... weather or not someone actually wants to track down whoever deleted/uploaded/modified that content, and what lengths they may go to within the confines of a warrant or the time/$$$ necessary to look, are big questions.
If there is an active investigation in-place, and you're considering tampering with evidence, you really should know that even if you've done nothing wrong, tampering with evidence makes you appear guilty more than anything else.