Is Samsung 300m worth buying in 2020?

Good decision buying the camera. It has much better IQ than most Canon APS-C cameras, including the new ones. Small, light-weight, articulating/tilting screen+good videos with NX 18-200mm lens.

1 - when my daughter moves I catch her hands in blur, probably have to experiment more with the shutter speed

Raise shutter speed and adjust ISO if needed. I have many pictures taken with IS0 3200 with NX300/300M and they are fine

2 - In head and shoulders type pictures my daughter appears with a larger head than in reality, honestly she is so delicate IRL, but in photos she appears like at least 1 month older.

Barrel distortion. You must be taking close up pictures at the wide end of focal length. Try distancing from the subject and taking pictures at the tele end of kit lens. There will be slight pincussion distortion but the head won't be bigger than desired

Probably have to step back and then crop?

Also bought hot shoe plastic covers, not sure if had a good deal, these might be universal, think I paid £4.5 for 3.

They are useless I lose them the moment I start using any camera

Now on another note and if you can help me here @Isolaverde I gave CeX a shot and ordered some cameras.

Ordered a Galaxy NX body - and got a white NX 2000 in shabby condition (their mistake, goes back)

Don't bother with other cameras than NX300/300M/500 and NX1. I have tried nearly all of them

Also ordered an NX500 which was listed with 16-50 in B condition listed as Black colour and received it in Brown colour. It came with a strap that doesn't have hooks and 2 extension tubes (10 and 16mm) Was thinking of keeping it and then selling the NX300m but:

- initial disappointment in the colour expected and delivered. I kinda like the brown leatherete in front but why did they go with the colour scheme at the back I don't know, even the plastic of the screen and bottom is brown... Takes a bit to warm up to it

Don't worry about the colour. Its the performance that matters. NX500 is a far better camera than NX300M. I would sell off NX300M and keep NX500 only.

- The lens is 16-50 Powerzoom - what I find strange is the noise of the barrell, if I shake the camera a bit up and down I hear the barrel of the lens moving and creating a sound. Is this normal?

Of course I can hear the other noises which I take are normal (electronic humming, the noise when focusing, the noise when zooming, and the gyro motor) from my research these are normal but wondering why the barrel moves inside the housing of the lens both when extended and closed, of course IRL I would not shake the camera like a pepper shaker but I am just wary that this unit might have been dropped.

This is a known problem with 16-50mm powerzoom lens. Not a tight fit of barrel in lens. Some times the barrel drops out of place and camera stops working as a result. I often had to tap on barrel to get it back in place for the zoom motor to work properly. Not a big issue and the lens is fine otherwise.