"My father is a taxi driver."
Yes, 父 (ちち) refers to your own father while お父さん refers to other people's father, giving respect to other's family
Depends who youre talking to.
My friends never say 父 in front of me...thats too formal. They tease me if I use the "masu" form of verbs for example.
But, to people outside of friends and family...yes...just like many other examples it depends on the context.
お父さん seems childish if you talk to people outside of the family. In front of such people you should refer to your father as 父 unless you're 3 or something.
Not just childish, but you don't use honorifics for your own family members. It's rude.
Anyone with this problem where all words are filled when in landscape mode on phone?
しゅ (written in kanji as 手, meaning "hand") is added to certain nouns to indicate someone who does that (the thing the noun describes). Hence うんてん (運転) "driving/operating (machinery)" --> うんてんしゅ (運転手) "driver".
Also happens in compound words such as せんしゅ (選手), meaning "player" (of a sport).
I assume you're talking about the しゅ in うんてんしゅ (運転手). The entire word means driver/chauffeur.
You can be talking about yourself with contezt, so why would you not allow 「お父さんはタクシーのうんてんしゅです。」?
お父さん is used to refer to someone else's father, so that sentence, in context, would be "Your father is a taxi driver." The better way would be simply to say 父。父 simply refers to you own father.