The display issue may be due to a broken Display Cable which is a known issue with these USB-C Apple laptops, but the display issue could be caused by a number of other things too. You can try closing the display, then slowly & carefully open it to less than 45 degrees to see whether you can see video on it which will indicate a bad display cable (will require replacement of the entire Display Assembly...expensive). You can also try connecting an external display, but getting video to it can sometimes be tricky when there is an issue with the built-in display.
The laptop is warm because it is powered on and running even though you cannot see anything on the built-in screen. You don't have to worry about the laptop catching fire here.
Keep in mind this particular model laptop...the MBPro 13" 2016 & 2017 non-touchbar model...has a lot of known design flaws & manufacturing issues as well as some lesser known issues. The known issues besides the Display and Keyboard, are battery issues with a small number of them, as well as SSD issues (I have found the SSDs in this particular model has a high rate of failure compared to other models). Lesser known issues is the design flaw of the USB-C ports which are integrated into the Logic Board which wear down quickly when connecting/disconnecting cables a lot which then cause the cable connections to become very loose & unreliable.
If you have never had the Keyboard or Battery replaced, then most likely the battery is wearing out as well and will need replaced at some time so keep this in mind when considering whether the repair cost is worthwhile. Personally I don't recommend putting money into this particular model beyond a battery replacement due to all the potential issues with this model.