Hi Jroyse,
I'm a relative newbie myself here.....and own a 2008 NER Whizzer which is a little bit different animal, so I defer to owners of WC1's and folks like Quenton. This thread would be more widely read if it was in the Whizzer section, I think.....
Can't answer all your questions cause I don't have the answers and am not going to steer you out to left field with some half baked answer! Anyway, the box is for the 12 volt battery that supplies the juice to power the light and horn........... Looks like if you are missing the battery, then most likely you are missing the battery cover (easily found on Ebay and the bolts that hold it in place..... Not sure if the air cleaner assembly is removed for the picture......at any rate, the NE-5 style air cleaner with the Quenton Guenther crankcase vent is a must for this model bike, versus the WC1 air cleaner that routes crankcase air into the air cleaner assemby and promptly clogging the filter with oil.....
Last year I was able to purchase a copy of the Whizzer dealer service manual from Debbie at Whizzer USA. I HIGHLY recommend this manual for you......the manual was written for the WC1 model Whizzer, of which your's is. In this service manual is a clear and concise electrical wiring diagram of your machine. If they are unable to come through for you with a copy of this manual, PM me with your address and I'd be more then happy to have a copy of mine made and sent off to you.
The Whizzer USA website has a breakdown of parts for this bike in their online parts catalog. I don't know what they stock for it, though.....I believe seeing your exhaust pipe there....possibly someone can post a pic of their WC1 exhaust set up......if not, I'm sure with all of the pics already posted here in past posts, there must be a a photo of your exhaust pipe bracket and tab.
And again, Quenton is always around, willing to render assistance as requested!
This bike looks to be a Whizzer knock off, possibly Cruzzer? I'm not seeing the Whizzer trademark name cast on the crankcase side cover. The WC1 cylinder, in particular, the valve seats are problematic AT BEST and will require significant work. Again, I have to steer you towards Mr Guenther for this fix....Either way, you should have a nice bike when the wiring is put back together........and all the required modifications are made.
As a final note, the Whizzer Newsletter is a must have for the Whizzer owner. Quentons' article, in this months newsletter, on the 1999 Whizzer crankcase mod is a must read...... I've owned my NER for over 3 years now and did without the newsletter......after subscribing and buying 2 years worth of back issues, I now realize how much info is available in that little publication. Well worth the price of admission!
Take Care,
Mike