The heating and engine temp should be consistent when running. There could be an issue with the system and id recommend getting it into a garage to be checked
Welcome to the forum you'll find the members on here are a friendly and helpful bunch. With any electrical problem with the car I'd recommend a diagnostic check to see if any fault codes flag up.
Could you take a picture of the box?
Welcome to the forum you'll find the members on here are a friendly and helpful bunch 🙂
Check YouTube for buyers guides. Check service history for proof of gearbox oil changes, cambelt and water pump changes etc
Welcome to the forum you'll find the members on here are a friendly and helpful bunch 🙂 I'd recommend a diagnostic check to see if any fault codes flag up. Are the auto lights working properly when driving?
Have a look to see if the gearbox has been serviced. Check to see if the cambelt and water pump has been changed. Make sure all electrical items work.
Check buyers guides on YouTube.
It's definitely headlight washer. This can be quite normal in these very cold conditions. Don't push them back in as you can break them. Let them go back in on their own once it's warmed up.