Big question with lots of opinions I'm sure. Here is mine.
1. MTBF is designed for non-repairable assets to determine their expected life. It is for population of assets so you can mathematically derive when best to take intervention to ensure a specific overall reliability across the population. This calculation can be utilized at the component level but must only be used per single failure mode of single component type. For example you can utilize MTBF for pump seals. This will give you some level of expectation of life expectancy give your current practices for installation, medium being pumped, and duty cycle. All things in this calculation should be same (same component, same operating context, etc).
2. MTTR is designed for repairable assets. This is the calculation you are looking for and want to utilize if the asset can be restored to base condition. When using this calculation, and given your example of finding an issue, the issue should be included. You are seeking to know how often you must visit the asset to take intervention outside of standard PM inspection. All corrective interventions regardless of if they are found prior to catastrophic failure should be included.
3. Yes, this a result of doing proactive inspections. However, the functional failure has occurred or you would not need to do the work. In your work categorization you should have, in SAP PM, a task type which identifies the work as found during a PM or condition monitoring work order. This will give you the effectiveness of your PM / condition monitoring program. If you never find issues your PMs are too often. If you always find issues your PMs too far apart. Rule of thumb I have read is a 4 or 5:1 ratio, meaning you should find a follow up corrective task for every 4 or 5 PMs executed.
4. As for your requirement for investigation I would say it is your criteria that needs to be looked at. If the event is recurring perhaps it deserves to be investigated but if it is a wear component you expect to fail over time perhaps not unless it is not reaching its life expectancy. A blanket statement which says all must be investigated seems a bit overkill. Maybe look at the trigger criteria and make adjustment if needed.
Again, opinions vary as do context of your asset base, the risk you can take, etc. This is just my input.