Is it true that English has no future tense?

Short answer: Yes, of course English has future tense ... for everyone except the most technical, and for them it doesn't have a future tense because they define "have a tense" in a non-intuitive way. So you can go ahead and say confidently that English has future tense.

Longer answer:

Most everybody (within the monolingual English speaking community) thinks of "I will do that" or "I am going to do that" are unequivocally future tense. It would be perverse to think otherwise; both those sentences describe a situation that occurs in the future. That's how most people, school teachers, newspaper editors, newspaper readers, most everybody thinks. And that is correct. For the majority of people.

However, language scholars, that very small set of expert thinkers who tend see a lot more theoretically of English and also of other languages, use that phrasing differently. "X has a Y tense" is technical terminology that means "The language X has verb forms that inflect for the Y tense". That is, the verb form itself is modified rather than having extra words around it to convey the tense. You have to use 'will' or 'going to'. English can express future tense grammatically (with 'will'), but technically it doesn't 'have' a future tense (assuming that what is meant is 'future tense inflections'). 'Will' is definitely a grammatical marker for future time. 'Going to' is less definitely a grammatical marker, it might be more accurately and technically referred to as a periphrastic future.

I think that a less misleading way for more pedantic people to express it would be to say that English has only two inflected tenses. That way both insiders and outsiders to the technical community of linguists would both know what is going on.

As to your specific examples, the way you label them doesn't really correspond to what anybody else says, informally or formally. I think you may be taking things too literally, or choosing one aspect of a definition out of the expected context.

I will do my homework tomorrow.

This is not present tense, it is future, whether 'tomorrow' is mentioned or not.

I am going to the bathroom.

Yes, This is present tense. Present continuous. You are in the middle of the action of traveling to the bathroom (or in AmE, it is a euphemism for in the middle of the action of micturating or defecating).

I am going to school tomorrow.

This is totally valid English grammar. The use of tomorrow changes the interpretation of 'going' from 'traveling' to the marker of the future. Or it could be interpreted as a contraction of "I am going to go to school", explicitly the future of 'traveling'

We are going to Italy in Spring

Totally correct usage. Ambiguous tense, but not ambiguous meaning.

We are going to Italy 2 years ago

Obviously wrong. Whether present or future, 2 years ago is contradictory, being in the past.

You might wonder why 'will' isn't considered to definitively say that "English has a future tense" (in the pedantic sense that is. In the less fully formal sense it totally says that English has a future tense). 'Will' is syntactically in the category of 'modal verbs', can may must should etc. Its meaning is certainly future time, but syntactically it is a helper (auxiliary) verb. But pedantically, 'will' is not grammatically about tense (I find this reasoning a bit of a stretch, but that's what being pedantic does).

To compare, French can express the future by either "Je verrai la peinture" (I will see the painting = I see-future the painting) or "Je vais voir la peinture" (I am-going to-see the painting). The first one is a true future tense by having an altered root and an inflection, the second expresses the future by a separate word (just like the English 'going to'). English doesn't have a corresponding inflected form for the future. So linguists say 'French has a future tense but English does not have a future tense'. That's just strange sounding technical language for linguists.

This might seem like cavilling (isn't all technical pedantic terminology there to cavil?) but it is useful terminology for language comparison. Some languages inflect for all tenses, and some don't have any inflections for any grammatical function.

Chinese, for example, is said to have no tense at all. You might wonder how they convey information. How do they wake up in the morning? How do they tie their shoes? Oh my god the Chinese are so inscrutable, with no sense of time they see past and future at once like those aliens in that movie and it's so beautiful and sad at the same time and etc etc

That is of course nonsense. What is the case is that verbs in Chinese (Mandarin) do not inflect for tense at all (or for anything), but they can just as easily express the time of something happening by using ... it's going seem stupid when I say it... it's so obvious... words for time. 'Now', 'next week', 'before my parents met'.

Also, people aren't idiots; context can help determine the sense. Suppose you can't hear how 'die' is conjugated in 'When my great-grandfather [die]' . Did that happen in the past, or in the future? If I am telling you this from my retirement home, it probably happened in the past. If I am visiting a nursing home, it's probably in the (near) future.

Here's a weird thing about English. Consider the sentence "I visit the park on Saturdays". There's no tense there, there's no inflection. But it means that you often go to the park in the past and will likely do so in the future. How do they know when exactly they're doing something? Is anything specified at all? How do they wake up in the morning? Oh my god the English are so inscrutable, how can they know anything, they are timeless and can see equally into the past and the future like time travelers it's so beautiful and sad at the same time etc etc

Strictly speaking, no Germanic language has a future tense, only a present and a preterite. All others are analytic tenses compounded with various auxiliaries. Since the present tense simple in English, unlike in other modern Germanic languages, is never used to describe what's happening right now, perhaps it would be better to call it the non-past. The progressive/continuous form has, for the most part, limited the simple present to 1) habitual/repeated actions, 2) general truths, conditions, and opinions — and, in conjunction with a temporal adverb, 3) future events.

Old English had no future tense and only the beginnings of an analytic one:

Mind that there was no Future tense in the Old English language, and the future action was expressed by the Present forms, just sometimes using verbs of modality, willan (lit. "to wish to do") or sculan (lit. "to have to do").

This parallels other ancient Germanic languages such as Old High German or Gothic: when translators into those languages encountered Latin or Greek future tenses, they either paraphrased or used the present tense.

The modal distinction between volition and agency is preserved in Middle English:

The future tense is formed much the way the future tense is formed in Modern English. The only difference is that shall (used as forms schullen, shallen, etc) and will (forms as willen, wellen, etc) had a difference in meaning. This difference is that will indicated desire or wish (much like High German Wollen), so saying I will go there was similar to I want to go there, and shall was involved indifference of the will, so saying I shall go is similar to I'm gonna go without desire to do so. The verb to go was not used to form the future tense, but the verb to be with an infinitive construction could have been.

Your thesis is that this modality still obtains today and that the so-called future tense is in fact not temporal but modal. If we were still speaking early Middle English, you'd likely be correct. In the transition to Modern English, however, modality began to yield to temporality. The only vestige of modality is the difference some British speakers make between I shall and I will, a nuance which has completely disappeared in American English.

The common way to form what is termed the future in English is will/shall + bare infinitive:

In 3.5 billion years, the sun will be 40 percent brighter than it is right now, which will cause the oceans to boil, the ice caps to permanently melt, and all water vapor in the atmosphere to be lost to space. Under these conditions, life as we know it will be unable to survive anywhere on the surface, and planet Earth will be fully transformed into another hot, dry world, just like Venus.

This description of the future end betrays not a trace of modality, but does strongly suggest that Modern English has a strictly temporal future tense, albeit with a former modal as auxiliary.