Is it possible to go from Probolinggo to Bromo on a motorbike?

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.

I'm a Probolinggonese and right now (June 2015) there's no place in Probolinggo that rents any motorbike or scooter to bromo.

It was possible in 2012-2014 as "lonely planet Indonesia" book (edition 2013) mentions on page 183. At that time, Sinar Harapan hotel provided motorbikes for rent. It was my friend and I's idea, but it's not working anymore.

I have lived in Malang close to Bromo for a year (2015-2016) and have heard of friends going on motorbike before that time, but more recently those roads were deliberately closed off so that people need to go on a more expensive jeep tour. The roads themselves are not great but doable, but you may run into trouble with the locals if you attempt this without really knowing which road to take. You certainly won't be allowed to take the main road up, it's gonna have to be an alternative small road.

Just to provide a possible alternative, I realize pricing may become outdated but last year in May 2016 me and my friends rented a jeep from Malang to Bromo, we managed to get it for 1.25M (excluding entrance fee), shared with 6 people. Roughly 200k rupiah each isn't too bad for a trip like that. However, this was after some intense negotiation in Indonesian language which we speak reasonably well, and only because we rented the jeep as a whole.