'93 Subaru Legacy - Check engine light, hard starting, & trouble codes,

How long did you have the battery disconnected? It should be disconnected a couple of minuets to cause the codes to erase. It it doesn’t, you know the procedure.
The engine computer thinks that the oxygen sensor and the evap. solenoid are inop, and the engine coolant temperature sensor has an abnormal signal. To be inoperative, they could be disconnected, or have poor connections. Disconnect, and reconnect, each electrical plug to the three components. Doing so, may wipe the connector pins and restore the connections.
You need to check for volts to each components, and check the out puts. 12 volts go the the ects. 5 volts to the oxygen sensor. Maybe 12 volts to the solenoid. A repair manual tells you how to do this good stuff. Another source is this site: http://www.autozone.com/shopping/repairGuide.htm?pageId=0900c15280066ea5