What level of damage to an aircraft is acceptable to let it depart?
Aircraft Maintenance Engineer here.

To evaluate if an aircraft is safe to fly we use three main reference documents:

1. MEL (minimum equipment list).

This document contains the list of elements that are allowed to be inoperative.

For example a pneumatic valve, a computer, a seat, etc...

All unserviceable elements will have to be fixed within a predetermined duration/limit.
(Usually 1, 3, 10 or 120 days)

If the failure is not fixed by the due date, the whole aircraft will not be released to fly. (No matter how small the fault is)

2. CDL (components deviation list).

This document contains the list of components that may be removed from the aircraft.

For example a damaged refueling service door can be removed, if the maintenance team is unable to fix it before departure.

These will usually be associated with performance penalties.

CDL defects will also come with a deadline.

3. SRM (structural repair manual).

This manual contains all information needed to evaluate structural damage such as dents, scratch, lightning strike.

This manual will also give you the information needed to evaluate if a damage is "no-go" (meaning the aircraft is not safe for flight) or if the damage is within tolerance.

If it is within tolerance you will also get time or aircraft cycles limitations (a cycle is a takeoff and landing)

Regarding your picture:

The plane will not be allowed to fly with a blown tyre. A blown/damaged/worn out tyre is a no-go.

However, airliners are designed to land safely with one blown tyre. Aircraft incorporate lots of redundancy so as to eliminate single points of failure &mdash; for example, most commercial jetliners are designed to be able to land safely even if they lose an engine. Efficiency, comfort, and ease of operation may be reduced but the situation can be kept under control and is still within the aircraft's design specifications.

I can see damage on the landing gear door. It is possible that an aircraft would be allowed to fly with the landing gear door removed if a fix cannot be performed before departure, however; the aircraft may experience increased drag. This would be covered in the CDL.

The aircraft should also receive extensive inspection before departure and after landing.