Question 1: How exactly a crosscap relates to a Mobius strip, a $^2$, and a Klein bottle?
I knew that
(1) Mobius strip glued the boundary with a disk $D^2$ is $^2$.
(2) Gluing the boundaries of two Mobius strip gives a Klein bottle
(3) Connecting sum two $^2$ gives $^2 \# ^2 $ = Klein bottle.
Euler characteristic $\chi$:
Mobius strip with $\chi=0$
Disk with $\chi=1$
$^2$ with $\chi=1$
Klein bottle with $\chi=0$
Question 2: Crosscap has $\chi=?$