I found a solution.

Although my question is not answered and I don't exactly understand WHY this problem happens, I want to share a simple summarization of the solution with you. I accepted that the Linux kernel simply doesn't provide support to Virtualbox after an update. They know the reason.

Anyway, the problem happens after almost every kernel update or package upgrade. If you try

$ sudo modprobe vboxdrv, you get modprobe: ERROR: could not insert 'vboxdrv': Required key not available.

The solution is to manually sign the key EVERY time this happens, because the signing is KERNEL-DEPENDENT. It is related to Ubuntu 16.04 WITHOUT disabling Secure Boot.

Steps:

1. Check if installed headers ($ dpkg -l | grep linux-headers) match the running kernel

   ($ uname -r).

2. You can always use this command to see a list of enrolled/signed keys:

   $ sudo keyctl list %:.system_keyring.

3. If you already signed a key for Virtualbox (with the old kernel), you need to remove it. There is a way to remove a single key, but I remove them all in once (all keys that I enrolled):

   $ sudo mokutil --reset. Choose a password and restart the computer.

4. You will be asked automatically by Ubuntu on booting. Choose "Reset MOK" and use the chosen password.
5. If you haven’t created X.509 key yet, create folder for the key and navigate to it.
6. Create the key with: $ openssl req -new -x509 -newkey rsa:2048 -keyout MOK.priv -outform DER -out MOK.der -nodes -days 36500 -subj "/CN=Akrog/". Instead of “MOK” and “Akrog” choose names (e.g. “vbX509Key” and “vbKey”).
7. Enroll the key with: $ sudo /usr/src/linux-headers-XXX-generic/scripts/sign-file sha256 ./MOK.priv ./MOK.der /lib/modules/XXX-generic/updates/dkms/vboxdrv.ko. Instead of “MOK” use the specified name and correct kernel version instead of "XXX".
8. Import the key with $ sudo mokutil --import MOK.der. Again, use the specified name instead of “MOK”. Choose a password and restart the computer.
9. You will be asked automatically by Ubuntu on booting. Choose "Enroll MOK" and use the chosen password.

Now you can normally start and use the Virtualbox.

I had the same problem with a Asus UX360UA i7-6500U (Intel® HD Graphics 520), ubuntu (with unity) boot on low graphic with the 4.10.0-37-generic and the intel-graphics-update-tool makes no updates.

My solution was simple, switch back to the 4.10.0-35-generic and wait for the next kernel....