Was Keener's character gay?

Don't think so. In one scene of the surveillance on Sylvia's apartment, Penn arrives to solo the night shift watch, relieving Keener and another officer who had been watching all day. The other officer leaves fairly quickly, but Keener lingers, saying to Penn - twice, if I recall correctly - that she can hang around "if you [Penn] want me to." The invitation was mildly expressed (i.e., it wasn't like she was openly vamping), but it caught my attention nonetheless.

I think Keener's character has romantic feelings for Penn's character; however, she has never overtly expressed them because Penn's marital circumstances (previously married, recently widowered) made it one of those "timing was never right" things that never had the chance to develop into a relationship - at least, not in the movie's timeline.