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.
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