Axis declares war on Ned and Belg triggering West Invaded and ending Appeasment.
Appeasement ends as soon as you declare war on the Western faction (i.e. as soon as the game starts in your case) as per 10.3.4.1.
Also West Invaded happens due to entering those countries, not right away when you declare war on them. Not that it makes much of a difference in practice.
Is Italy immediately and automatically at war with Germany as a consequence of being activated as a West ally? Can Italy then cross the mountains into Germany, as appeasement is over?
Yes and yes. Your feeling is correct.