What I can do to [get] the flash working?
Take it out of slave mode.
The only mode which is working is the wireless one where flash is triggered by in-camera flash.
This means that your flash is in a slave mode. That tells the flash to ignore any signals on the hotshoe, and to listen to either an optical sensor (if it's an optical slave mode) or the internal RF receiver (if it's in a radio slave mode on a flash that has this; your Neewer doesn't).
When you try to use a flash on a camera hotshoe, it has to be set to "listen" for signals on the hotshoe, and that means being out of any off-camera slave modes.
Is OM-D compatible with any hot-shoe flash or not?
It is and it isn't. It's fully compatible with any four-thirds flash if you want TTL/HSS and camera/menu communication.
But it's also partly compatible with any ISO-compatible flash if all you need is to fire the flash in sync with the shutter. But with this type of operation, you can't do anything else from the camera. You have to manually set the power level, you can't use TTL or HSS, or change any settings on the flash via the camera menus. It has much more limited usefulness.
As far as I can tell, that Neewer model only comes in Nikon and Canon flavors, so on a micro four-thirds camera, you won't get anything but firing in sync, because the camera hotshoe isn't speaking the same language (electronic signal protocol) the flash understands.
Which flash should I get if not?
If you want full TTL compatibility, you need to look for a flash that's compatible with Olympus/Panasonic (e.g., an FL-600R/FL-360L [same flash, made by Panasonic, rebranded by Olympus] or a third-party flash (e.g., Godox, Nissin, Metz, Meike, etc.) that does Olympus/Panasonic TTL.
Also it's interesting that in the moment I connect flash into hot shoe of my camera, it fires. Is this normal?
If both the camera and the flash are on, yes, this can happen. The firing signal is a simple short between sync and ground. As you slide the flash in, the sync pin can hit a "live" connection, and that will fire the flash. You should have the flash turned off when you put it onto the hotshoe, and only turn it on after it's been secured in place.