Set of stairs leads top of building. End of stairs there is a 5 metre walk to the front door of apartment. Should there be a fire door in between top of stairs and front door?

 Also opposite the front door is a final exit door that could can lead to a horizontal escape to next buildings terrace. This door has no push bar but a key in a glass box.

Alsoooo...The stairs don't have any manual or automatic ventilation, is this place in need of some work???

The key in a glass box is not acceptable and there are other acceptable methods, not necessarily a push bar.  It sounds like a conversion and I would survey it in accordance with  HOUSING – FIRE SAFETY Guidance on fire safety provisions for certain types of existing housing.