No.. Mr Sheen is harmless. It has a bit of silicone wax in it so it'll help the water run off the paint and give a nice shine!
At least for a few days, after when it will have evaporated away.
Some people polish their whole car with it.. but I think that's a bit eccentric. Use proper car-valeting products.
The minute scratches (swirling) are probably minor enough to be reduced or even eliminated using Autoglym Super Resin Polish.
When you're in the Big H. buying the SRP, buy a couple of the small foam applicator pads (e.g. Meguiars). Apply the SRP to the marks using a damp pad and polish away. Light to moderate pressure will do it. You don't have to get rid of them completely.. they'll vanish over time, if you keep on it. SRP also contains fillers that will mask them for a while.
Next time you wash the car (search for car washing advice.. you'll wonder how it could be so complicated!
), use the SRP and the pad to "polish" the whole car. You won't need to press too hard, as your paint is new but if you have any bonded on contaminants (paint feels like grit is stuck to it) look at clay (search for "clay"... yes.. you are turning into an anorak!
When you're done buffing that SRP off...(use an old terry towel or a Microfibre cloth) .. apply a coat of wax. This protects the surface better, hides some minor marks and also makes the beast easier to wash next time.
Wax is a pub-discussion topic. If you want to start "basic" you can do worse than the Simoniz or Turtle Wax paste wax in a tin (the second foam pad comes in handy for that) from the Big H. while you read up on all the different varieties of waxes and other potions available.
Read Polished Bliss - Car Care Products - Car Care Advice - Detailing Services - Scotland - UK
to see what's available and how to use it... but at the very least you should own:
1 x bucket (2 buckets is better.. see "two-bucket method")
1 x Car shampoo
1 x Wash mitt (sheepskin or Microfibre) Sponge if you really want to make the fellas on here cringe...
1 x Drying towel
2 x applicator pads
Super Resin polish
Microfibre (Terry will do) towel (for buffing)
You can optionally own a clay bar kit (use it after washing, before polishing) but your paint is new so you have time to build up to that.