Car wouldn't start then it start after I unplug battery for 1 minute

I had a strange problem on my 2001 MK4 Golf. I had a good battery with enough voltage and when I turned the key, the glow plug light was working and all lights were working in the dashboard, everything seemed normal but the starter wouldn't turn. It did a click noise and then it stopped doing anything. Finally, I was able to start it by unplugging the battery for one minute. I figured it did a reset on something related to the immobilizer... Now, what should I do to make sure it won't happen again? My remote control isn't working anymore (even when I change the small battery inside it) but I was unlocking the door with the key until it refused to start...

The immobilizer will not prevent the starter from turning; it will allow your engine to start, then kill it after 2 seconds or so.

This sounds more as if it is a bad starter solenoid or starter, or bad battery connection (that you made better taking the battery terminals off and on.) The click leads me to suppose that the interlock circuit and ignition key are working properly.

Starter is almost brand new (it's an OEM Bosch starter not a crappy Hitachi), I'm just going to assume it was caused by a bad battery connection. I'm going to tighten a bit more the nut on the battery terminal. Is there anything else I could do to make sure the connection is strong enough?

Clean it with a battery terminal brush before you tighten it.
Put on some of those red and green chemically treated felt washers.
Coat the terminals with silicone grease, or vaseline.

You can also have a bad connection to the lugs on the ends of the wires.
Or the braided cable built onto the starter can degrade - corrode and fall apart.

So watch for hot spots and/or voltage drops across terminals as you crank the car. Bad connections will get warm.