let's me take your example itself
f=x and g=x+1 and c=2
So x< 2 (x+1) here also it is true f(n) < c.g(n) (here f should be strictly smaller ,equals condition not possible)
and f=x and g=x+1 and c=1/2
x > 1/2(x+1) here also it is true f(n) > c.g(n) (here f should be strictly greater ,equals condition not possible)
So i'm not getting where your are finding the intersection between these 2
for the second question
O( g(n) ) ∩ Ω( g(n) ) here it can't be empty always
for example f=x and g= 2x
so for big-oh x < = 1/2(2x) (where c=1/2) -----> x<= x
for big-omega x>= 1/2(2x) ----> x>=x so there is a possibility of intersection