Weird problem #2 sounds like it could be a bad thermo/time switch. 
this switch completes the electrical circuit for and limits the duration of 
the cold start injector.  An internal bimetallic strip, heated both 
electrically and by engine heat, determines the cold start injector on time by 
providing a short duration electrical ground. It could be dumping too much 
fuel into the engine when it doesn't actually need it. Check all of the other 
engine sensors, including the mass airflow sensor, the fuel pressure and the 
injectors themselves. But by all means, KEEP THE FUEL INJECTION.  It's really 
much easier to live with than carbs.

Weird problem #1 is the bane of all aircooled vee-dubs.  It is caused by the 
resistance of the wire and connectors from the ignition switch to the starter 
solenoid.  The resistance gets higher as it gets warmer.  Also as things age, 
the resistance goes up and there is not enough current left to pull in the 
solenoid.  Make sure all the connections are clean.  Then install a universal 
starter solenoid in the area so that it routes power from the main battery 
positive on the starter to the ignition switch connection on the original 
solenoid.  The ignition switch wire itself goes to the new solenoid which 
requires much less current to operate.

Hope this helps,

Steve,

Kirbyst@a1.rpr.rpna.com

'77,'81 buses
'67,'70,'74 bugs
'68 1602,'79 528I bimmers