Hi Cris,
don´t worry about not finding the neutral gear if you stand at a traffic light! Just move the bike forward about 10- 30 cm and try to enter the neutral in a sensitave way when mooving. It will enter smoothy

Soon you will get used to enter the neutral in the last meter before stopping. If once you stand still without mooving the bike it´s not that unusual to have the problem described by you.
The little bit of oil will not be the problem of the clutch. Be aware if you try to loose the special nuts of the clutch with a fissured srew driver. Sometimes they are veeeery tight an there is a certain risk that the stud bolts get loose in the basket. I don´t know how your situation of spareparts is in NZ but this center baskets are quite rare to find in good condition. If you can untie the nuts without problems clean al the discs . there is one disc concav (o convex) This shoud be the first steel plate to enter. You don´t have to use much force to tight the nuts, they can not get loose by themselves! And allways mount them with copper pate, so you can desmount it if necesary. If you fear to brake the stud bolts you can give all the amount a good bath of brake cleaner an simply let it dry. But I don´t think that your problem is an oily clutch, perhas more that old oil sticks the plate together, if the motor didn´t work for a long time. But after 700 km that should be done.
Before opening the clutch first be sure that the cable and all that stuff it right (not much bended/forced, working easily, clearence ...
If you desmount the whole clutch, have a look at the center basket. There will be marks from the steel plates. If they are deep, smooth it a little with a file. Behind the center basket there is a special washer withe 4 ears. They can be of plasic/nylon or brass (or something like that) The plastic one is better!
Good luck with that signorina!!!
norbert