This is not normal, I am quite an expert with samsung phones.
This is more likely a hardware error.
By the way when you hold volume up+home+power buttons, how long do you hold it for? The correct method is to hold volume up AND home buttons together first, after that hold down the power button. Don't let go of any button untill you see the phone's logo. This usually leads you to the recovery screen. If you can't even get this to work, I suspect the boot is corrupted.
Try this to get download mode on your phone. Hold volume down + home buttons first, then keep holding the power button for few seconds untill you see a warning and options to continue/cancel.
If you successfully got to this download mode, then you can flash a new rom yourself using ODIN. Here's the link to the page:
http://www.modifyandroid.com/stock-roms/samsung/how-to-flash-stock-rom-for-samsung-galaxy-s5-mini-sm-g800f/
NOTE: If you can't get to the download mode also, then it is definitely a hardware fault, most likely a bricked device!