My Chinese teacher gave me the name 海追远 last year. I picked out the surname 海 because it sounds like my last name, he just gave me the first name 海追远. Now that I've started a college Chinese class, my teacher told me 海 is an uncommon surname and that it doesn't fit with 追远。 I'm looking for honest opinions on how common 海 is as a surname and whether it works with 追远.
For your first question, there are about 48000 people whose surname is 海 in China, accounting for about 0.004% of Chinese total population. (This data comes from https://wenda.so.com/q/1591524650610065.) It is really a rare surname that I have not seen anyone surnamed 海 around me.
For your second question, I do not know why your teacher told you it doesn't fit with 追远. I think it is ok. And if you still mind it, 海明远 mentioned in comments looks more proper.
Furthermore, having a rare surname is a cool thing in my eyes. When I was young, I always dream to have a rare or compound surname, like 慕容 欧阳 姬. But different from you, I can not choose my surname by myself. I think your teacher may just be worried about your possible misunderstanding of Chinese Name System. It should not be related to name itself.
Is 海追远 "weird" sounding?
Not weird, but certainly "unusual" For males, the traditional names tend to connote qualities like "intelligence", "strength", "bravery", "patriotism","wealth", etc., qualities all parents wish for their children to have.
追远, "to chase far", as a name may elicit a teasing response like 追什么远? (what are you chasing thus far?)
So, it is not "weird", in the negative sense of the word, but I wouldn't name my child that, unless I live on a remote, isolated island and it is my fervent hope that with 海追远 my child would one day escape away from the isolated loneliness and "chase / seek his fame and fortune in far off lands over the seas"
Or, (speculatively), your teacher thought that you have a long, long way to go in your Chinese language studies and it thus amounted to an "ocean of a chase of a distant objective"