Was Buffy ever friends with Cordelia?

I would say Cordy and Willow became friends but not Cordy and Buffy.

Yes. Of course it depends on what you're calling friendship. Even if they aren't super-best friends they certainly became average friends at some point. Not at the very start of the series but certainly so when Cordelia was dating Xander. They hung out a lot together and Cordy was a part of the Scooby Gang whether she (and the others) liked it or not.

As you imply in your post, there is a lot of feeling going on "behind the scenes". Cordelia acts like a bitch because she wants to make clear she thinks of herself as "superior" and Buffy kept her distance to make a point she wasn't a vain person like Cordelia (like she had been once in the past). These demeanors aside, both of them were more often than not, there for each other to talk to and for support in times of need, even if those came with the mandatory snarky comment from the other side.

Again, I'd like to point out they weren't really compatible or super intimate but friends? Sure why not. We all have had this friend or another who wasn't really part of our group and we made fun of but that doesn't mean we wouldn't lend a hand if he asked for our help. Right?

Cordelia is arguably the character with the most internal development in Buffy and Angel. I'd really like to know what it would be like if they started hanging out again on Sunnydale after all she went through up to the final of the Angel series. There'd probably be a huge "elephant in the room" because.. you know, Angel?

To a degree, yes. The two were very similar in their early lives (Buffy before being called and Cordelia in the early seasons) and they quickly bonded through that in the beginning of season 1.

Buffy and Cordelia were friends but there were some pre-existing things that kept them from being super close. See "Homecoming". They were too competitive and superficial over a lot of things, mainly because of Buffy's old personality butting heads with Cordelia. Cordy was around a lot because of Xander but I don't think Buffy ever saw her as a valuable addition to the group. Buffy tolerated Cordelia and Cordelia tolerated Buffy, but they never had a lot of meaningful moments to bond. They were friends but it was a minimal/frayed friendship. I think, in the end, neither were too upset about losing the other after high school but Cordelia clearly expressed sadness over Buffy dying a few years later.

That being said, I think Buffy was closer with Cordelia than she ever was with Anya. I never felt like Buffy and Anya had any kind of relationship outside of Anya being around because of her relationship with Xander. Without Xander, Buffy and Cordy still had some interactions.

What remains to be seen is Buffy learning that Cordelia and Angel fell in mutual love. That would clearly put a wedge in any sort of relationship the two would have.

Wesley, why can't I stay?

Cordy had such a strange attitude. She's just like Anya, but I'd argue that Anya is way more tolerable and a evident friend to Buffy. Cordy is way too busy being Cordy. She hides her emotions well, but I'm sure she cares about Buffy.

"I wish I wasn't afraid all the time, but I am."
-V for Vendetta

I'd say Buffy was more friends with Cordelia than she ever was with Anya. There was some mutual respect between them - even if they didn't get along

I think so yes, read a great fanfic once where after becoming part demon CC calls Buffy and tells her that now she understands what its' like to be a freak.