Q. I notice that the preposition "at" seems to mean "try to do something" and we don't know if it succeeds or not.
Let's look at your examples
"he shot the cat": the cat is dead or injured BUT we are not sure if he meant to shoot the cat? -- "he shot at the cat": we are not sure the bullet hit the cat or not Correct but we do know "that he intended" to shoot the cat
"he grabbed her hand": her hand was in his hand However we do not know if he intended to grab her hand or arm or? -- "he grabbed at her hand": we are not sure if he succeeds Correct but we do know "he intended to" hold her hand
The bird and biting examples follow the same logic. However the "Tea one" does not work, it would have to be "at his tea".
A. A preposition is used as a relationship word. It is a word that can explain the place, time, or relationship of a sentence. Therefore in your examples it does not mean "try to do something." The use of "at" indicates the intended action and we are clear about what the intended action was. However the use of "at" coupled with the context of your examples does show what "I/He/They/It were trying to do"
intend; verb; to have as a plan or purpose
The Preposition "At"
So, the word at is a preposition. A preposition is, in a sense, a connector. When you use the word at in a sentence, you should notice that it smoothly connects the other words around it.
If the word "at" isn’t working as a connector, then it will most likely be working as an adverb. As such, rather than connect words it will modify one of the words in the sentence.
A preposition is a relationship word (hence a "connector"). It’s the sort of word that can explain the place, time, or relationship of a sentence, and link the words together. In the case of at, this preposition is commonly used to express specific location (e.g., “Joe is waiting at home”), specific time (e.g., “The appointment is at 9 P.M.”), or a specific point (e.g., “The progress meter is at 75 percent.”).
Correct Usage of "At"
Believe it or not, there are just two points to remember about the use of prepositions like at.
You need to choose the right preposition for each sentence, remember a preposition is a connector bringing the parts of the sentence together. A preposition like at is likely to follow certain words in order to make the relationship crystal clear between other words in the sentence (looking at, depends on, driving by).
Here’s an example sentence:
I am looking at him. In this example, you could only use the preposition at. Otherwise, the meaning of the sentence would be compromised and completely unclear. “I am looking from him” or “I am looking on him” would make no sense.
Position in a Sentence
Next, consider how and where the preposition at is placed in a sentence, that is, a preposition must generally be followed by a noun or pronoun. The noun or pronoun counts as the object of the preposition. This can get a little tricky, but if you look at the above sentence again, you'll see what we mean:
I am looking at him. The analysis would go like so: the preposition at is followed by the pronoun him.
Ref Correct use of "at"