Can my company do a Reg D offering at the same time we're raising a crowdfunding Reg CF offering?
Yes you can do a Reg D offering concurrently with a Reg CF offering. However, you'd want to make the Reg D a 506(c), which permits general solicitation, because while concurrent offerings under Reg CF and 506(b) are theoretically possible, it can get very complicated, especially because in the CF you are going to have to disclose the existence of the Reg D offering (and vice versa).
What, if anything, do the regulations say about deal terms of concurrent Reg D and Reg CF offerings?
e.g. Can an issuer offer a different (cheaper) price per share to Reg D investors, offer warrants, different voting rights, or other "benefits" to incentivize them for writing larger checks than the ongoing Reg CF investors are getting?
No regs really, other than the integration rules in Rule 152.
If the different offerings have different terms then you want to make sure the dilutive effect of the "better" deal is fully disclosed in the other offering.
But better terms in the Reg D offering than the Reg CF offering is pretty common.
it's generally a matter of making full and complete disclosure.