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.
The constructions you are quoting are of the should have + past participle type.
(wrong) It should have came down.
Here, the verb to come is used in its preterite ("simple past") form, which is wrong.
(right) It should have come down.
Here, the verb to come assumes its past participle form, as it should after should have.
The same is true with could have took (wrong) and could have taken (right). In short, you were right in both cases.
Why? Because the modal verb should and the verb have play auxiliary roles (they are "auxiliary verbs"). The carry some information, and the verb coming after them is left to carry the rest.
The auxiliary verb have already tells us that the action is in the past, so we don't need the simple past form of the verb to come. We need to attach a "weaker" verb after the have, a verb that merely carries the information about the kind of the action (to come) and not about the time of the action. Otherwise, we would have a "much of a muchness" situation.
The Past Participle is such "weaker form": a non-finite verb.
Best Answer
You are using the Present Perfect in your first sentence, but you'd need a duration phrase to make it more natural for this context:
Without that duration phrase, the reader might think that "at some period in his life he has lain on the ground, but now he's quite an active guy". This would be unfitting for a statue.
In the second sentence, I'd delete down:
With down, it might be understood that he is in the process of lying down: he hasn't yet settled in his position.
The third sentence is unfit because laid is the past-tense form of lay, not of lie. "Lay" means to "place something somewhere". It's a transitive verb. That's how we usually use "lay":
It's quite hard not to get tangled in the diverse forms of "lay" and "lie". Here's a related question:
The ELU StackExchange even has a special tag to mark questions related to the usage of lay and lie.
Here's a handy table with forms of lie and lay:
(from "Lie Lay Practice Worksheet", Freeology.com, licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 license)
Read also: