Learn Hardcore Japanese: kyou ha situmon ga hutatu arimasu.

は would make 質問 the topic (As for the questions, there are two), which is possible but not the most natural choice when you just want to say there are/you have questions.
が is the usual particle with ある / いる when stating existence.

Why not を?

を marks direct objects of action verbs (eat, read, do, etc.).
ある is not an action, it is an existence/possession verb, so it pairs with が, not を.

What does あります literally mean? Is it to have or to exist?

Literally, あります is the polite form of ある, which means to exist / to be (there) for inanimate things.

本があります。 – There is a book. / I have a book.
時間があります。 – There is time. / I have time.

So:

質問が二つあります。 – There are two questions. / I have two questions.

English often uses have, but Japanese is using an existence verb. The idea is “two questions exist (for me / right now)”.

Why do we use あります instead of います?

Japanese has two main existence verbs:

あります (from ある) – for inanimate things: objects, ideas, events, etc.
います (from いる) – for animate things: people, animals.

質問 (questions) is an inanimate concept, so you must use あります:

質問が二つあります。 – There are two questions.
学生が二人います。 – There are two students.

Is あります present or future tense here? Does it mean I have or I will have?

Japanese polite non-past form (あります) covers both present and future. The exact meaning depends on context.

In this sentence:

If said at the start of class: 今日は質問が二つあります。 – Today I have two questions (I plan to ask them now).
If said while introducing an agenda: 今日は質問が二つあります。 – Today I will have two questions (for you to answer later).

So grammatically it is just non-past, and context decides whether English uses have or will have.

Why is 二つ used instead of just 二? What is 二つ exactly?

二つ (ふたつ) is one of the native Japanese counters (ひとつ, ふたつ, みっつ, etc.) used to count general objects when you do not use a more specific counter.

一つ – one (thing)
二つ – two (things)
三つ – three (things)

In 質問が二つあります。, 二つ = two (items) and it is implicitly understood as two questions.

You could use a more specific counter:

質問が二問あります。 – There are two questions. (問 is a question/problem counter.)

In everyday speech, using 二つ for questions is common and sounds natural.

Could I say 二つの質問があります。 instead? Is that different?

Yes, you can say 二つの質問があります。 and it is grammatically correct. The nuance is slightly different in structure:

質問が二つあります。 – Questions (subject)
- quantity (two)
  - exist.
二つの質問があります。 – Two questions (as a noun phrase) + exist.

In practice, the meaning is almost the same: There are two questions / I have two questions.

However, 質問が二つあります。 is more basic and is usually taught first; it feels slightly more neutral and natural in many contexts. 二つの質問があります。 can feel a bit more like “There are two questions (as a set/itemized thing).”

Why is there no particle after 二つ? Should it be 二つが or 二つを?

二つ is acting as a counter directly attached to 質問, so it does not need its own particle here.

The structure is:

質問が – questions (subject)
二つ – (are) two (in number)
あります – exist

The が already belongs to 質問. 二つ is just modifying 質問 by telling you how many.

You could change the structure and then add a particle:

二つの質問があります。 – There are two questions.
質問を二つしました。 – I asked two questions.

But in the original sentence, 質問が二つあります。, 二つ is directly tied to 質問が, so no extra particle is needed.

How is 今日 read, and why is it きょう instead of something like こんにち?

In this sentence, 今日 is read きょう.

今日 has several possible readings in Japanese, mainly:

きょう – today (the most common, everyday reading)
こんにち – nowadays / this day and age (more formal or set expressions)

In 今日は質問が二つあります。, the meaning is today, so the correct reading is きょう:

きょうはしつもんがふたつあります。

How polite is this sentence? Could I use it with a teacher or in business?

The sentence is in polite form because of あります.

Politeness levels:

質問が二つある。 – Plain/informal.
質問が二つあります。 – Polite (standard polite speech).
質問が二つございます。 – Very polite / honorific style (business/formal service).

今日は質問が二つあります。 is appropriate and natural:

To a teacher
To coworkers
In most everyday polite situations

If you wanted to be extra-formal (e.g., in a presentation), you might say:

本日は質問が二つございます。 – As for today, there are two questions. (very polite)