AUDI A4 1.8T automatic 2008

So this is my first time here as I’ve only just bought a second hand Audi A4

On purchase I was made aware the car had new bearings and an oil pump replaced and that the car only starts up on 2nd or 3rd attempt.

This is the problem that I have:

1 – The car will not start at all on the first attempt. It swings and swings.
second time around it takes after 2 swings. some times second time will also just swing. then the 3rd time it takes. This happens weather cold or hot. morning or day.. then she purrs like a kitten on the road, high or low speed.

2 – On acceleration and only on acceleration the car vibrates but still performs very well. Feels like she over revs at times not sure. Like a slipping clutch on a manual.

Not sure if this is of importance but the right indicator bulb needs replacing and also the left rear park light. this comes up on dash when ignition is on and it complains about this sometimes while driving.

I’m hoping someone can point me in the right direction. I’m no mechanic but I’ve always done my own services and minor repairs on my cars.

Anyone have any ideas please? Will be greatly appreciated