Bitcoind sync process I accidentally deleted all the files that the standard bitcoin client stores to disk, except blk*.dat files. When I run bitcoind, he write that he doing sync. This sync is very very slow. What does bitcoind doing with this blk*.dat files while sync? And why there is need to connect to the internet during sync (it did not work without internet)?
The block files are indexed (the blocks/index subdirectory), and if you just delete the index, bitcoind will assume you have no blocks and start over downloading from the network.

You could instead have used the -reindex option to scan for blocks on disk and rebuild the index. This may still work, but likely won't as you've already started downloading again, and overwriting the existing block files.