Can Ryougi Shiki see "Death" in inanimate objects?
It depends on what you mean by "inanimate." 

Generally, intangible objects like words cannot be "killed" because they do not exist. A phenomenon like a storm cannot be killed, but the particular event (rain/wind/clouds) can be dispersed. 

In the Tsukihime Dokuhon Plus Period book, Nasu makes mention that:


  Ryougi's Shiki Mystic Eyes is superior to Tohno's. She (unlike Tohno) is able to perceive the death of just about anything, however she is limited to what she perceives as "alive." 


So take something like a chair, which is seen as "alive," because it's not broken. Ryougi would see lines on a new chair, but not on a broken chair, because she believes that it is already "dead." 

Note that this notion only applies to her perception of the "living" as opposed to if something "has life." Even though Kirie Fujou's ghosts were "dead," they could still be killed because they are "alive" in the sense that they could interact and interfere with the real world, as if they were alive.

As a side note, Araya's barrier is linked to his body, so when Ryougi cut it, it's like cutting a part of Araya (this is why he is in pain).