I'd like to send and receive faxes on my computer, and wonder if the iPhone can be connected to the computer via a modem so I can use it as a telephone (for telephone-based stuff, not for internet or whatever)?
No, that's the phone sharing its internet connection. The iPhone doesn't do this, but other phones do.
He's asking about sharing the PHONE connection with the computer to use as a fax line. No cell phone in the world does this.
What the original poster needs is a service that takes faxes and turns them to e-mails and then sends to them to you. I've never used any of these, so I'm not suggesting them. I'm just reporting what Google gave me: