I have been to Bromo a couple years back (2011). We took a tour bus there and a public bus back however.
If memory serves me right, it's a proper road (for Indonesian standards), with a lot of potholes, but some sort of solid covering. Not well maintained, but definitely doable with a scooter if it's not your first time riding one. There are also a lot of locals around with scooters at Lawang. I found one picture online, which shows you the top of the road at Lawang, just before you'd go down into the crater, this should give you an idea on what kind of cover to expect.
Note that I would still recommend you to walk from that point onwards, as I am not sure how easy it will be to ride your bike on the ash inside the outer crater towards Mt. Bromo which is a around 20 minutes walk inside the crater. Another popular spot is the viewing spot, slightly up a hill on the side where you can get a view as seen on the picture below. This part is usually done either on foot or horses, and I don't think that part will be doable on a scooter.
t Bromo from view point, taken by myself
I can't really help you for the night part, as I don't know what will make it safe enough for you. If you want to see the sunrise though, I'd suggest to travel in the night before and stay in Lawang overnight, so that you can hike up to the viewing point early in the morning and thus also avoid the crowd.
Note that if you don't want a tour but aren't too keen on the scooter, you should be able to board one of the mini-buses at the terminal in Probolinggo and then another one down again from Lawang.