Starting issue could just be the battery, have it checked.
The door issue is common on all VWs for the last 15 years or so. The locks have gremlins, yours probably has a broken wire/poor connection from the micro switch which monitors the position of the latch. So it sees the door as open all the time, it won't lock when the door is open.
If you are handy with a soldering iron then pull out the tools or get a new lock otherwise.
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