Я жду тебя у входа в библиотеку.

The verb is ждать (to wait (for)). In the 1st person singular present tense it conjugates as я жду. Many Russian verbs don’t form the 1st person singular by simply adding -ю to the stem; ждать → жду is an irregular-looking but standard form.

Ты is the subject form (you). Here, you is the person being waited for, i.e., the direct object, so Russian uses the accusative case: ты → тебя.
So я жду тебя = I’m waiting for you.

No. Russian ждать normally takes a direct object without a preposition: ждать кого/что (to wait for someone/something). That’s why it’s жду тебя, not жду для тебя or similar.

у is a preposition meaning by/near/at (someone’s place or something) and it requires the genitive case.
у входа literally means by the entrance (standing near it).
Other options exist but can shift the nuance:

• у входа = near the entrance (very common)
• на входе = at the entrance area/at the point of entry (often sounds like describing a location in a system/building layout)
• в(о) входе is generally not used for this meaning

The dictionary form is вход (entrance). After у, Russian uses the genitive case:

• вход → входа
  So у входа = by the entrance.

Because в + accusative usually answers where to? (motion/direction): into/to the library.
Here it’s not motion in the main verb (I’m waiting), but вход is an entrance into something, so Russian treats it as the entrance leading into the library: вход в библиотеку.

Compare:

• вход в библиотеку = the entrance into the library (which entrance it is)
• в библиотеке = in the library (location)

Yes, it’s essentially a noun phrase: вход в библиотеку = the entrance to/into the library.
Structure:

• вход (main noun)
• в библиотеку (prepositional phrase modifying вход, telling what it leads into)

Russian has no articles (a/an/the). The sentence can mean either an entrance or the entrance depending on context. Often, in real situations it’s understood as the entrance (the relevant one), but Russian doesn’t mark that with a separate word.

It can be omitted if the context makes it clear, because the verb ending -у in жду already shows I.

• Я жду тебя... is explicit and can add emphasis/clarity.
• Жду тебя... sounds very natural, especially in messages.

Default neutral order is subject–verb–object: Я жду тебя.
But Russian word order is flexible for emphasis:

• Тебя я жду... = It’s you I’m waiting for (contrast/emphasis)
• Я тебя жду... is also common and slightly more conversational

Russian present tense often covers both I wait (habitual) and I’m waiting (right now). Context decides. In this sentence, with a specific place (у входа...), it usually means I’m waiting (right now).

Yes, common additions are:

• Я жду тебя у входа в библиотеку сейчас. = I’m waiting for you at the entrance to the library now.
• Я уже жду тебя у входа в библиотеку. = I’m already waiting for you at the entrance to the library.

Yes, тебя corresponds to ты (informal you). Formal/plural you is вы, and the accusative is вас:

• Я жду вас у входа в библиотеку. = I’m waiting for you (formal/plural) at the entrance to the library.

As written, у входа в библиотеку is most naturally understood as by the entrance to the library (typically outside or right at the doorway). If you specifically mean inside the library, you might say something like:

• Я жду тебя у входа, внутри библиотеки. = I’m waiting for you by the entrance, inside the library.
• Я жду тебя в библиотеке, у входа. = I’m waiting for you in the library, by the entrance.

Yes, it’s singular. The dictionary form is библиотека. After в for direction, it takes the accusative:

• библиотека → библиотеку