Problem is this "H" thing. It's not really a disk. I don't know what it is.
At first I thought it looked odd that the device shows up right at root in Device Manager, but it turns out that's normal, it's how all removable disks appear in there. (These items represent a portable device for Windows Media Player – that's a separate object from the "disk" itself.)
Is this an SD card reader?
Most likely yes. Front-panel card readers on desktop PCs are almost always USB-attached. (In laptops they may be either USB-based or directly PCI-based.)
It is completely normal that an empty reader shows up as an empty drive, because it is indeed still connected (it's the same as how all other fixed-reader, removable-media drives work).
(Although not all card readers work that way – some of them might completely disappear from the USB bus when the card is removed.)
Is there a chance to hide it?
In Device Manager, right-click "USB Mass Storage Device" and select "Disable device".
Alternatively, open the PC case and just unplug the reader.
I've checked this reader with one card. It's working. But after I safely remove the card I found out that there is no possibility to read the card again. If I'm putting the card inside the reader and there is no reaction. Nothing. Whatsmore, this icon in the bottom right corner had dissapeard and in the device manager section there is one problem with USB connection.
The card reader might be broken, or just improperly connected to the motherboard.
I would test it through a Linux "live CD" (e.g. from Ubuntu) – open terminal, run
dmesg -ewL, insert and remove the card a few times. If it shows the same problem there, then you can be certain it's a hardware issue.
On the other hand, if it works on Linux, then the reader may need a special driver (although that would be very unusual)... It might, however, be a general problem with the USB 3.0 drivers.