Sorry for the late response, but here's my shot at 'answering' this.

A few things before I do: 1. this is an era and subject in which archaeological and written evidence is scarce and rare, you must always know that there is a lot that may have happened that we simply cannot know until more work is done to dig things up, so to speak; and 2. Though things are vague, there are at least a few conclusions you can make from the record. For example, if all history was wiped out until 2017, and then historians in 4000 CE looked back on 2017, they would not assume that all civilization was spontaneously generated if they had reason to believe there was a cause for losing history on dates before that. Likewise, that Vedic influence showed up in the record at 1500 BCE and in that location means that the development of Vedic culture existed, for who knows how long, before that moment--this could be a few years to thousands of years, and we can't say much about geographical spread though.

So, to answer your first part: Since the Indus-Saraswati civilization ended around, roughly, 1900 B.C.E., we might be able to speculate on some Vedic presence there (or nearby) around that time, but the Vedic period in that region certainly began at latest by 1500 B.C.E., so yes, they would have (historians unanimously agree that the Rg Veda was written 1500 B.C.E., which is where we begin the 'Vedic period' classification).

The nature of the Vedas, the ancient scriptures of which the Rg Veda is the 'first,' is incredibly rooted in oral tradition, so its possible that the Rishis who supposedly codified it and determined its structure of oral preservation (which consists of an intense system of memorizing every word, syllable, and sound, backwards and forwards, in every other word, and multiple other formats) preceded the historical Vedic era. Beyond the Kush/Punjab/Sindh/Gangetic Plane regions, Mittani seems to be the only place during this era of an obviously Vedic influence because of those scriptures--but, since its the only evidence, we know only of scriptural influence on those who wrote the texts, and definitely can't make conclusions about Indic people's societal role in general or how it got there.

Side note: before any evident Vedic influence, the Indus civ. itself had ports for trade in dozens of cities. My point being, the world was likely more interconnected connected during these ancient times than we even realize or have evidence for, though we cannot speculate on specific trade routes or anything without reason.

Finally, as for the Levant, here is why we 'know' that there was Vedic influence as much as we can know something like that. There are many correlations between the Vedic pantheon and Vedic Sanskrit with the gods and languages of all sorts of people, the great many number of places including those in today's "middle east," western and eastern Europe, and South Asia. BUT, those all point to cousin-like relationships, as per the PIE hypothesis I'm sure you know--and Anatolia has tons of this influence, and is incredibly rich in its own right (some even identify PIE origins as Anatolia, but thats contested and fallen less in favor). In Mittani, for one, the gods of these scriptures in question are clearly, not even vaguely, Vedic gods, and not just cousin-relationships that we see when equating Vedic gods with other pantheons. Iranian developments that descended from indo-european history, while the closest branch or 'offshoot' from the proto-indo-iranian peoples along with Vedic people (indo-aryans being proto-indo-iranians), even shows a relatively notable departure between the iranian from the Vedic, but these Mittani documents are unquestionably Vedic when it comes to Varuna, Mitra, Indra, Nasatya, and Agni. Beyond that, the names of multiple people are almost exactly the same (Priyamedha, Pritasva, Tvesaratha, Indrota, Artasmara, Pritasva), the stories concerned are equivalent, and the subject matter and narrative is exactly Vedic (horse-training; horse-rituals; both names and the commentary/guides). These people are of course the of the kings of Mitanni, and there are a few examples of kings who ruled over the Kassites where the names are sanskrit-like. This only applies for one or two that are evident, wheras the Mitanni names have obvious Sanskrit definitions and are related to deities or concepts like Rta (Artaatama, as per the Amarna letters, Rta--Dhaman, "his abode is Rta." There are a bunch of better examples, I can give the names and translations for those if you'd like).

So, in sum, the number of words that are not from the Hittite language and are instead Vedic is relatively substantial. Kikkuli, the "master horse trainer"'s texts are especially interesting. According to the "Catalogue des Textes Hittites" (CTH 284) which takes a look at the monographs, there are also lots of number-related Vedic loanwords, and many phrases are interrupted with an abrupt use of a Vedic (sanskrit)-like word, followed by an explanation of that word, as if there is no Hittite-equivalent word, he was forced to go back to his native tongue, which is followed by a brief explanation of what it means. It's almost like an admission that there is some difficulty in translation from Vedic to Hittite and attempting at reconciling the possible loss of concept in the process. That's about all we know! This would mean either that both the Mitanni ruling kings (and some Kassite kings) and the Vedic peoples of India held an incredible retention of ancestry in language and religion that did not differ much from its roots in stark contrast with other the other descendants of PIE or proto-proto-indo-aryan people--or, somehow, there was either a transmission of ideas, or a transmission of people.

Take from that what you will.