Can the base of the plant be in shade?

Can.....doesn't need to be and full sun is just fine.

As to whether or not your vine will stay confined to the trellis, eventually no. Blanche is a big robust vine and will easily outgrow the trellis. But Honeysuckles really have no method of attachment that would allow them to grip the garage wall but they could very easily climb to the roof and continue on from there.

This will not happen immediately however. All vines take time to establish and develop their full potential. If you keep your honeysuckle pruned you should be able to keep it confined to the appropriate space.

There is a general Vine forum, btw :-)