Bmw 330 has burning rubber smell when going up a steep hill. It happens when the auto changes down a gear and the smell goes when the car reaches the top.

No
not if you drive it accordingly
steep hills are best negotiated in low fears with plenty of rpm applied
for example on a steep grade with a manual 1st or 2nd or possibly 3rd and the rpm around 2000 constant will cause it no harm
automatics will go to first automatically so keep the rpm at that mark
engine damage is experienced when the gear selected is too high and the engine is placed under extreme load so down shift to allow the engine to work easily
When going down hill select first and only use the brakes to maintain the 2000 rpm range and allow the engine compression to hold the car speed back
the rule of thumb for experienced drivers is this
you select the same gear to go down that it took to come up

This sounds like an oil leak. Look for leaks at valve covers, and other areas above the exhaust. It often is the burning off of this small volume of oil, that makes the smell like you describe. Inspect the whole exhaust...Something is melting, somewhere.

Possibly TPS starting to fail. but first have oil level & quality checked

This sounds like you may correct in saying a misfire. this sounds like a bad coil that looses power under a load. Is check engine light on?
If the light is on or not go to an auto parts store like auto zone.At auto zone they will scan the computer free of charge and give you a print out of any codes. the computer will store code or codes when it detects a problem. A bad coil will make it store a code or should.
see if there are any misfire codes, this code will help pin point the bad coil. if there are codes write me back and give me the code. I think this car has coil over plug ignition system. this just means instead of sparkplug wires the coil sits right on the top of the plug.
Also replace spark plugs if it has been a while since they have been replaced. check air filter too. a matter of fact if you have over 70,000 miles on car tune it up. plugs,furl filter if it has one, air filter and pvc valve and clean throtle body.
Okay, just checked it out, this car if a 1.6 engine then does have a coil but not on spark plug, rather coil with sparkplug wires going to plugs. try replacing spark plug wires and plugs first and if problem remains then replace the coil. let me know the results. good day

auto or manual?
Either way you need the box repairing or replacing, which ever is cheapest. Gearboxes are not for amateurs and VW are known for gearbox issues.
the battery thing is likely to be coincidental

1) Check the brake pads all round. If a caliper is sticking (not retracting fully) then the pad will remain in contact with the disc rotor and gets very hot giving off burning smells. 2) If you have a manual transmission it could also be that your clutch is at the point of failing. A slipping clutch is indicated in gear the engine races irrespective of the the car accelerating along with a smell of burning.

have you changed your tran fluid
if not check and see if it is burned or has a burned smell to it
have the pan dropped and change fluid first

It sounds like differential lube. gear lube.A little bit go's a long way.check the seals pinon and axles.the cover on the differential look at the vent too.also gear lube on brakes it's not good.some times you have to pull tires off to see the oil.

Does the temp gauge go up too?