Samtools how many reads were mapped as a pair

[Fay.huo]

Hey, I am new to bioinformatics. I get stuck on the question How many reads were mapped as a pair.
I have created a .bam file that only contains mapped reads. Then, I need to figure out three questions.
How many reads were mapped?
How many reads were mapped as a pair?
How many reads were mapped as a "proper" pair?
I can solve the 1st and 3rd question, but I am confusing about the 2nd question. After searching on google, I have found the following code:
samtools view -F 0x40 filename.bam | cut -f1 | sort | uniq | wc -l
samtools view -f 0x40 -F 0x4 filename.bam | cut -f1 | sort | uniq | wc -l #left mate
samtools view -f 0x80 -F 0x4 filename.bam | cut -f1 | sort | uniq | wc -l #right mate
source from :https://gist.github.com/davfre/8596159
I don't sure it works or not. Can anyone help me?

Also, I need to find the Arabidopsis thaliana gene that associated to AT3G13276 and identify how many inserts mapped to that region. Does this need to use IGV to solve it?
Thanks.