Isn't DHCP only the way my router gives me IP?
No, it's also the way your router gives you extra information – gateway IP, DNS server IP, default domain name, time server IP, et cetera.
Why I can connect from Live USB to Internet normally, while in chroot I have to add new DNS?
Linux stores the DNS server address in a file. Since chroot alters how programs see file paths, it means they'll find a different file at the
/etc/resolv.conf path, which means they'll possibly find a different address stored there.
(Your DHCP client doesn't know anything about your chroots, and only updates the main
/etc/resolv.conf file – it cannot search and update all possible files.)
A common workaround is to link (bind) the "inside" path so that it refers to the same "outside" file. For example, you can do this before chrooting to /mnt:
mount --bind /etc/resolv.conf /mnt/etc/resolv.conf
(Of course, later you'll have to unmount this before the system will let you unmount /mnt.)