When someone says "He is smarter than I and she put together," what is the function of the phrase "put together"? Is it considered an adjective?
Yes, contemporary traditional grammar would call this a "reduced adjective clause." Of course, it's not an adjective at all. It's a verb phrase functioning as a modifier.
It's worth parsing and reducing the sentence.
X is smarter than Y.
X = He. Y = I and she put together*
'put together' is a verb phrase (derived from the verb, 'put') logically grouping 'she' and 'I'.**
although I concur with Blessed Geek that it should be 'she and I put together'. * that is why the suggestion that 'me and her put together' is wrong... because neither is the object of 'is', but rather the passive object of 'put'.