I can't say whether your trigger is eligible or not - only the platform can decide that. Why don't you try it?
As to why you haven't got coverage of the test class, when you run the test it will tell you which lines aren't covered. As this is a test class, that means you aren't executing the lines and should probably think about getting rid of them.
That smacks that you have code in your test class that isn't being executed. Quite unusual IMO.
what is the IMO ?
Is my trigger is eligible for deployment?
IMO = in my opinion.
I can't say whether your trigger is eligible or not - only the platform can decide that. Why don't you try it?
As to why you haven't got coverage of the test class, when you run the test it will tell you which lines aren't covered. As this is a test class, that means you aren't executing the lines and should probably think about getting rid of them.