Tor is fully secure only if I use HTTPS, right? But not every site supports it, so I had an idea - some online anonymizers support HTTPS, so if I connect via Tor (Tor -> anonymyzer -> site), will that be the same?
In that case the connection between anonymizer and webserver would be just as unencrypted as the connection between exit node and webserver. It would still be possible to eavesdrop on it.
All you get from this is that you get an additional layer in your onion circuit (the outer layer is the https connection between you and the anonymizer) and that you shift your trust-relation from your exit-node to the anonymizer service.
There is simply no way to communicate fully-encrypted with a communication-partner who doesn't support encryption. No matter what you do, the last hop must always be unencrypted in that case.
No and No to both of your questions. In reference to your first question google TOR DNS leak just as an example. While this is simply an example a good question to pose to yourself is "how secure is acceptable?"
In reference to your second question, in this situation you would simply moving one end-point of the encrypted traffic from your computer to a third-party computer. While this would encrypt your traffic over the TOR network it inherently places someone in the middle of your traffic with full access to unencrypted data.
You should use something like that: https://www.eff.org/https-everywhere It's created in colloboration with Tor developers, so you can rely on that if you already decided to trust Tor itself.
Edited: My bad. Yes, SSL Everywhere addon won't provide you secure connection in case the target website doesn't support SSL at all, thanks to raz for pointing this out. Then some free vpn service will probably be the most affordable and easy to use solution - http://www.vpnbook.com/freevpn or http://www.vpngate.net/en/, to name a few. You can force your vpn connection to be established on top of tor's connection. Please see this link for clarification: https://airvpn.org/tor/. They also offer custom vpn client able to do the trick out of the box. There how to configure this in linux (haven't found anything for ms windows): https://www.whonix.org/wiki/Tunnel_Proxy_or_SSH_or_VPN_through_Tor