SSH tunnels aren't VPNs
First and foremost, let me say say that you should install OpenVPN (or similar) on your VPS and save you a lot of headaches. When connected to a VPN, you can easily make sure that all traffic will get routed through it.
Normally, DNS requests do not get routed through the proxy. Your ISP is (supposed to be) your DNS server, and there's no need for it.
You can specify a local address as your DNS server (system wide), e.g.,
127.0.0.3 and set up port forwarding to a suitable DNS server:
ssh -p 443 -D 8080 -L 127.0.0.3:53:188.8.131.52:53 -f -C -N root@ip
Some applications also let you specify this. In Firefox, e.g., all you have to do is enter
/etc/ssh/sshd_config, change the line
sshd from sending data to the system log.
However, other processes might still log your IP address. For example, lastlog and last both store IP addresses you logged in from (using SSH is asically logging in)
If you're really that worried about logs, you can erase all files in
/var/log periodically. But keep in mind that if somebody gets access to your VPS, he probably can get access to the purchase information as well.