How do native speakers of Chinese pronounce Latin characters? I hear that "Windows XP" is commonly called "Chā P" because the letter "X" may be pronounced "Chā".

Are there other Latin characters that have similarly peculiar pronunciations?
There is no single way to pronounce a given letter in the Latin alphabet. The set of rules varies wildly depending on the speaker's geographical location. 

That being said however, I have, a few years ago, compiled a list of common (more Northern I believe) pronunciations of Roman letters. You can find it on this subpage of my Wiktionary homepage: Roman Letters in Mandarin.

For your convenience, I am going to paste it here:

字母  漢語拼音標音                    同音字
A     ēi    
B     bì                            閉
C     sēi                           塞（白讀）
D     dì                            地
E     yì                            意
F     ái fu                         癌副
G     jì                            既
H     éi chi    
I     ài 或 ái                      愛 或 癌
J     zhèi                          這（口語）
K     kèi   
L     ái lur                        癌露兒
M     ái mu                         癌木
N     ēn                            恩
O     ōu                            歐
P     pì                            僻
Q     kiùr  
R     àr 或 ár                       二
S     ái si                         癌四
T     tì                            替
U     yōu                           幽
V     wēi（聲母多為唇齒音 IPA: /ʋ/）    微
W     dá bu liu                     達不六
X     ái ke si                      癌克四
Y     wài（聲母多為雙唇音 IPA: /w/）    外
Z     zèi


Just on your specific example, chā is certainly possible as X resembles the cross. I have never heard of people pronouncing XP like this before though.