How can I explore my Android from my computer? I'd like to explore the content of the internal memory. I have a rooted phone with a terminal emulator which allows me to navigate through it.
The problem is that it's quite hard to use (small screen, no physical keyboard) and I cannot transfer files from internal memory to my computer to examine it (maybe memory -&gt; SD and then SD -&gt; computer but not the easiest).
Is there a way to have an SSH daemon (I've ConnectBot but it's only a client) on the phone, control it with a USB cable (with USB debugging maybe), install a backdoor (maybe no), or something like this?
The Android Debug Bridge (adb) can do pretty much everything you want. It is part of the Android SDK, just download the SDK and enable USB debugging on your phone.

You can get a shell with the command

adb shell


and you can get files from the phone with

adb pull


All the commands are documented in the first link I gave. 

Another way to get any file from the phone would be to use any of the backup utilities available and back up your whole phone to your computer where you can examine it. As you already rooted your phone you might already have a backup utility installed, or you can easily install one, e.g. Titanium Backup Root or Nandroid. I have no experience with those two though, I can't tell you anything more about them. 

The kernel I used to root my phone included ClockworkMod Recovery with a backup program already integrated, so you might already have that available too.