Modern Vespa

I have a 2002 et2. Several days ago I noticed strong petrol smell around bike and a few days ago petrol was leaking and dripping out below engine.
1. Today I replaced the fuel pipe. When I pulled the old pipe off the petcock outlet petrol came out continuously....should it come out?
2. After putting all back together bike started ok and idled ok but after 10 seconds or so the engine started to race?
3. When I go check on or start the bike every 3,4 days the throttle grip is stuck but frees with a bit of force. Would this be due to unlubricated or broken throttle cable or what would actually be sticking? I have oiled the cable.

1. Most modern fuel taps shouldnt flow continuously. They have a diaphragm built into them that closes shut so that fuel can't drip/ pollute. The diaphragm is pulled open by a vacuum line which runs from the tap to the manifold in front of the carb. Your system have 2 hoses coming off the tap? Make sure both are intact and pliable.

2. A racing engine indicates an air leak somewhere. Could be from bumping the carb in the manifold or the snorkel from the air filter. Make sure everything is seated, tight and aligned.

3. The cable should run free but it actually is 3 cables in 1. 1 from the throttle to a middle section, and then 2 from that (1 to carb, 1 to the oil pump). Disconnect the cable at the carburetor so only the oil pump and handle are connected in the Y. In 3-4 days, see if you still feel resistance. If so, your problem is in the upper segments. If not, youve narrowed it down to the carb.

Thanks for your advice

Is it better to get a new petcock instead of dismantling the old one?
Also re air leak:
Should there be some sort of clamp or clips on each end of the air filter rubber pipe, ie at carb and at air filter.
The rubber gaiter/hose that goes off the top of the air filter housing to a plastic air duct, would that cause an air leak issue? Mine has a hole in it.

I'd replace the lot - all those fuel and vacuum pipes, all the clips (use Jubilee clips, not the cheapo ones) and the petcock.

You need to replace the petcock.

Regarding the airbox, no - the rubber duct running up from the box is just to ensure the air going into the box is coming from through the frame duct and not straight up from the road (including puddle splashes etc). So it's good to have that intact, but not really a mixture issue.

The closes thing to air leak behavior to expect from the airbox is if the pipe from the box to the carb is loose, or if the box is cracked open, or if the filter is missing - these type of scenarios will allow air to flow more freely into the carb than it is jetted for stock, because the box and its filter (should be lightly oiled) are restrictive.

For air leaks you should check the attachment of the carb to the cylinder - make sure it's seated well on the carb and on the reed valve block; might need to refresh the seal there especially.