My first thought on question 1 was "Is it remapped?" and low and behold, it is
Let's start with the easy stuff then. Mayo in the filler cap = normal. It's the coldest part of a running engine and is where moisture in the PCV gases condensate. Happens to pretty much everyone over the colder months.
If you're not losing any coolant, you can skip straight past any leaking head gasket thoughts.
HPFP, I would get that sorted sooner rather than later and purchasing VCDS is also a good idea if you haven't got it already. These engines need a lot of fuel when remapped and the standard one can't cope in many cases, edition 30 especially.
The more complex stuff. DSG. I have a manual so have very limited knowledge, but get it checked out and get a DSG map done to remove the auto upshift when in manual mode and perhaps increase the clutch pack clamping pressure as APR maps are quite aggressive normally. It's quite normal to get the DSG harmonized to the remapped engine. You're probably not realizing the car's full potential!
And finally back to number 1. Mine is also remapped and I get the same issue when cruising in the 2000-2500 region. The lower the gear, the jerkier it can feel. It also jolts when coming off / on the throttle which is bloody irritating when trying to maintain a set speed in urban areas. I don't remember the standard map being like that.
It doesn't sound like there is anything drastic with your car to be honest. These cars can be a bit irritable when remapped at times.