Does Avast Home for Linux detect spyware?
Yes it does. Just not on Linux, of course. 8)
Yes, the avast for linux uses the same engine as all avast products and thus have the same abilities such as spyware detections 
Al968
Yes, the differences in detection abilities between windows and linux engine is nearly none - only few rare exe-unpackers won’t be decompressed (but this doesn’t mean that the malware won’t be detected, in many cases, we catch the packed form of the malware too).
regards,
pc