Copper comes from Old English, where it was spelled coper. It was probably spelled with two p's in Early Modern English because it was a short vowel, and there was a tendency to double consonants after a short vowel.
Proper comes from French propre, and before then from Latin proprius. It was probably spelled with one p in Early Modern English because there was a tendency to stick closely to Latin spellings (e.g. debt, which never was pronounced with a /b/ in English, but which had the 'b' added in Early Modern English because it originally came from Latin debitum.)
As the comments noted, all of the others come from a one-syllable word with the suffix -er added. The rule (simplified) is to double the consonant after a short vowel when you add the suffix -er.