Learn English – “has nothing to do with” vs. “does not have anything to do with”

Let's look at a few more sentences in the same tense (present perfect) as your first example:

Mary has eaten the cake.

I have finished the report.

Someone has taken my phone.

In each case the first phrase of the sentence is the doer of the action (Mary does the eating, I do the finishing, someone does the taking.)

From this it is clear that your first example does not make sense (although it is grammatically correct). An idea cannot do the deleting. It does make sense, however, to say: Someone has deleted the idea (actually, deleted the file would be a better example).

Your second sentence, on the other hand, is both grammatical and makes sense. It is in the passive form of the same tense. If we convert the examples above to the passive, then we get:

The cake has been eaten (by Mary).

The report has been finished (by me).

My phone has been taken (by someone).

Your second sentence fits in here:

The file has been deleted (by somebody).

We use the passive like this when we want to shift the focus of the sentence away from the doer of the action. Maybe we don't know who did the action, or it is obvious, or we don't care who did it. The passive allows us to focus on what happened and does not require us to mention the doer.

I think that "not a thing" is used for specifically referring to an actual set of items, whereas in your example it is not.

In these sentences, I have replaced "nothing" with "not a thing".

Correct: Not a thing was stolen when I left the shop unlocked.

Correct: Not a thing is preventing someone from getting in.

Incorrect: I've installed a new lock for you. Don't thank me; it was not a thing.

I feel that "nothing" in the context of politely refusing praise cannot be replaced by "not a thing".

Side note:

"not a thing" can be expanded to "not a single a thing" and "not even a single thing" for emphasis.