Or the 西 (xi1) in 东西 (dong1xi0) is pronounced with no tone?
No. It does not always apply.
xi0 might be another way of writing the neutral tone. I haven't seen it before.
For 东西, dong1xi1 (east and west) and dong1xi(thing) have different meanings.
To make sure whether a word ends with a neutral tone, without hearing how natives are using it, you'll have to look it up in a dictionary.
One cause of neutralization of tone is when a meaning is repeated. This is why 爸爸 is ba4ba0, for example. In the case of 喜欢，both syllables basically mean the same thing, so the second one becomes neutral. Same as 东西：even though they are opposites, in this context they just refer to "stuff over in that direction", so they are essentially repeating the same meaning as well. This is why if you pronounce the second syllable's tone as 1st tone, they you get the meaning of East-West: you have signaled that you actually mean them as opposites. 衣服 is another example, though you will find many examples of this.