I'm a US citizen with Canadian permanent resident card. My US passport is being renewed. Can I cross into the US with copies of both my passport and the passport renewal in my possession?
Yes you can. In actual fact it is possible to enter the USA with no ID whatsoever particularly if you are coming by land (airlines and ships will turn you away without ID).
However be ready to be detained while they thoroughly confirm you are a US citizen. See Does a verified US citizen need a police & medical report to return from Ghana?
Once confirmed though, USA immigration law gives all citizens the right to enter.
The Fifth Circuit in William Worthy, Jr. v. US, 328 F.2d 386 (5th Cir. 1964):
We think it is inherent in the concept of citizenship that the citizen, when absent from the country to which he owes allegiance, has a right to return, again to set foot on its soil.
See the real life response from this woman who lost her passport in Mexico. Vehicle customs, lost passport