Solid state relay voltage current rating
My question - if I wanted to control 12v @ 1A DC through this SSR,
  will it be able to handle it?


No, The limit is 110mA for DC, you can't go beyond 110mA DC without risking damage to the part.

There are other conditions that you can go over the 110mA limit, and the bottom graph shows you why, the switch is anywhere from 10Ω to 16Ω 





It appears to me that the switch would be useful for isolation and a switch for triacs or relays or but not for any substantial load.