Mac – White screen for 10.12.2 upgrade on mac mini

This answer assumes you have access to another Mac and a way to connect your hard drive to it.

Take out the drive of your Mac using this iFixit guide.

Connect to the drive and boot into Recovery (CMD + R) on the other Mac. Then, follow these instructions from an Apple help article:

Install from OS X Recovery

To install OS X on an external drive using OS X Recovery, follow these steps.

1. Start up your Mac. Immediately after you hear the startup sound, press and hold the Command+R on your keyboard.

2. After you see an Apple logo or globe appear, you can release the keys. If you're prompted to select a Wi-Fi network, select a network that has a working Internet connection.

3. When the Recovery screen appears, make sure you're connected to the Internet. If not, select a Wi-Fi network with a working Internet connection from the Wi-Fi menu.

4. Select the option to Install OS X from the Recovery window.

5. Follow the onscreen instructions to complete the installation. When you're asked to select a drive for installation, click the button "Show All Drives". Be sure you select the external storage device you've connected for this purpose, not your computer's built-in startup drive.

Your Mac might restart one or more times during installation. After installation is finished, your Mac automatically starts up from the external drive.

When you're done using your external disk, select a startup disk from System Preferences to have your Mac to start up from its built-in startup disk or another drive.

_{Your computer's serial number is sent to Apple as part of this process to help authenticate your request to download and install OS X.}

Once the installation of OS X on the Mac mini's disk is complete, insert it back into the Mac mini and follow the iFixit article's instructions in reverse order.

It sounds like a lot of the speed problems you're having comes from running newer software, macOS 10.12, that usually requires more system resources on a computer that has insufficient RAM (2 GB just does not cut it practically) and a slower HDD. However I don't think that would normally cause quite the amount of speed reduction as you have reported.

Nevertheless, for a quick speed boost, I would definitely recommend upgrading to 8 GB of RAM. Apple has information on how to upgrade, just make sure that the RAM you get is compatible with Apple's specifications. For example an 8 GB RAM kit from Crucial should work.

Additionally, I do know that HDDs can be a speed bottle neck, especially as they get more full and are older. You can upgrade to an SSD (here's a guide by Crucial), however that is a far more complicated process, and you will also need to transfer all your software from the old HDD.

But your speeds are very slow, so there is probably some software problem. You can try to diagnose that by going to System Preferences > Users & Groups > Login Items and see if you have programs loading at boot. Also if you can check Activity Monitor, that may show if something is using a lot of your CPU. In the end, you may need to do a backup, wipe your hard drive, reinstall macOS or an older version of OS X, and transfer your documents and settings.