I brought my puppy home around 1.5 months back.
This 4 month old puppy is constantly jumping and "nipping" on my adult dog.
Is this normal? A way of play?
And how long should i be expecting this behaviour to last before they both settle down?
Also recently, my puppy's been shaking her head a lot and wanting to scratch one side of her ear.
Is this something i should be worrying about?
After 6 weeks or so your pup should have settled down somewhat by now. The nipping, biting and rough-play that you mention is usually discouraged by the older dog pretty quickly.
However, it's nearly always best to let the dogs sort things out by themselves, unless you are really worried that one of them is going to get hurt. You can find lots of tips and advice on this on my Introducing A New Puppy page.
As for her shaking her head and scratching at her ear - I would definitely recommend letting the veterinarian take a look at that ear. It sounds as though she may have an ear infection or ear mites. Dog Allergies can cause ear infections, so that may be worth investigating.
Hope this helps. Best of luck with your pups.