There seem to be different conventions for headlines because they are not always complete sentences. For example, many newspapers would be fine with "Man killed by unknown assailant" as a headline, or "Budget increase passed by legislature", which have the same grammatical structure as "Apple betrayed by its own law firm".

Edit: oops, but that's not what this headline did. Nonetheless, the headline seems reasonable to me as a native speaker.

In that case, it should read "Apple betrayed by own law firm", leaving out the "its"

Trump elected by narrow margin

Lincoln assassinated by John Wilkes Booth

There's an implied "to be" verb, generally "was". This is very common in headlines.

The comma introduces a parenthetical phrase describing Apple, implying Apple is going to do something:

Apple, betrayed by its own law firm, countersues with gusto

Apple, ["who was"] betrayed by its own law firm, believes the best revenge is living well

You can. It would be similar to saying "Alice ate the cake." See the structure: subject subject verb object

Not the same structure at all. "Apple betrayed by its own law firm" is passive voice; your example is active voice. Passive voice in English is constructed with "to be" as the conjugated "helping verb" and the past participle of the actual verb. This is also why the law firm must be prefaced with "by"; the actor is not included in passive voice. earlyriser is correct in his proposed insertion of "was".

The equivalent to your example would be, "Law firm betrayed Apple, its client."