the topic of linearizing PDEs is very broad. You can linearize many things: the original PDE, a perturbated PDE, the original PDE is weak form, the discretized PDE...
Convective terms can be neglected when Re<<1, but I see no advantage of doing that; indeed, in a flow where viscous terms are dominant one does not expect numerical problems. In the opposite case (Re>>1) , the story is totally different.