Is it doable and safe to bike from Arras to the Vimy monument and if so, is there somewhere to rent bikes?
Google Maps will show you three routes from the Arras train station to the Vimy memorial, estimating around 50 minutes to cover about 14 miles. It points out a restricted road but I think that's just the access into the memorial grounds. Can't help about bike rentals myself, but Google can. Here is one source, from the tourism authority, an office that in general I have found quite helpful:
http://www.explorearras.com/en/outdoor-leisure-activities.html
You'll go through Neuville St Vaast first before reaching Vimy.