In your first video it sounds like it is low on compression. Do a compression test. This would be a good thing to do anyway on a car that you just purchased with little to no maintenance history. Easy to do test and could tell a lot. Also, check coolant temperature sensor. If it is getting a false reading, the ECU will not deliver the appropriate amount of fuel on start up. I just repaired a Volvo with similar symptoms where the temperature sensor would read fine and then all of a sudden drop to -147 degrees Fahrenheit. Needless to say, it was causing the car to run all sorts of crazy.