How often do I need to change the synthetic oil in a 2003 jetta tdi? and what grade of oil is best?

No, but you will see a small increase in fuel economy if you do. Plus, you can extend your oil change intervals with synthetic oil.

PLEASE DO NOT add any oil to the fuel,use redexe/wynns diesel additive to clean the lines/injectors/turbo vanes,but burnt oil will stick to the turbo vanes and could cause failure.....just use 10/40 semi synthetic oil as normal and change regularly.also let the car tickover for a min or two after a HOT run....

start wih #1,try using a can of redexe/wynns injector cleaner(1 can to 1/2 tank diesel)works well...#2 use 10/40 semi synthetic(best you can afford).i prefer not to use any other fuel additives(other than redexe)as VW group products are susceptical to "clogging"(dirty/high sulphur fuel dont help).i use a can of red every 25k,tailpipe dark grey......

don't know about the MPG it could be a lot of things. I have a 2001 Jetta TDI. You should NEVER change the oil at 3K miles. VW recommends 10K miles for a reason. The Synthetic oil has additives to remove varnish and it does that but it needs some time to replace the varnish and properly protect sensitive engine parts. 3K is oil changes are not a good thing for a TDI......Just use the right oil and change at 10K

using a high grade full synthetic is the best.
the oil grade which is 10w30 is for average driving conditions.
for winter,5w30 and 10w40 for summer is also recommended by alot of car manufacturers.The oil filler cap will tell you the oil grade.

There's a few that you can use - but in general

Use Full Synthetic only for the TDIs...

There's various schools of thought after that in terms of 5x 30, 5w 40, even 0w 30 or 0w40

I've been using Castrol Synthetic 5w 30. The original specs for my 2000 Jetta TDI may have been for 5w40 oil formulated for diesels - but with the Synthetics now I think the 5w30 is felt to be the best in terms of what works best in the Turbo

You can probably go every 5,000 to 7,500 miles between changes - a good thing since the synthetic oils are more expensive

by the way, some also suggest changing the fuel filter at every oil change. I'm thinking that's a bit much but others may disagree... maybe ever 15,000 miles for me

5w-30 is the correct weight of oil for your vehicle. If you are currently using synthetic oil then you must stay with synthetic. If you are using regular conventional oil, then you can choose to stay with it or switch to synthetic oil. Good luck

I looked it up, add 4.5 quarts oil and filter

There's no tappets adjustment needed for this engine as they are hydraulic lifters. Try using 10w40 instead and see what happens or otherwise try using Moreys oil treatment with your current oil as it seems to me that the noise is more at startup and for a short period. Cheers.

Sounds like it could be a sludge issue which can be caused from using the incorrect oil. It is a very specific synthetic type of oil that VW suggests and when you don't use it the engine can actually 'gunk' up and cause turbo failure. (This problem happened when a non-vw dealer sold me my car with the wrong type of synthetic in it and my turbo seized) 
Google: "1.8t specific oil"