There is no evidence that Rāmā was a historical person, though there is little evidence to counter that either. The claims made above are, frankly, absurd. I see not a single thing that can be called completely correct. It is full of half truths and outright fabrications. I'm going to debunk each claim in turn:

1. The Lion man is a spectacular statue, one of the first and finest examples of figurative art in the world, but beyond that we know next to nothing. Is it male or female? Both? Neither? Is it a God, a Spirit, a Human with a mask? We don't know. What we do know is that Viṣṇu as a concept did not exist yet, especially the Daśāvatāra Narasiṃha, which we can date to the Gupta period or slightly before during the composition of the Puraṇas. Hinduism, or more properly the cults that now make up Hinduism are descended from Vedic religion, itself descended from Proto-Indo-European religion. The PIE culture began to develop some 8,000 years ago in the Pontic Steppe before splitting off into various descendants who we now recognize. The Indo-Aryans mad their way to India sometime in the 15th or 16th century BCE and spread until they were the dominant culture in North India, bringing their religion along with them. Viṣṇu appears in the later mandalas of the Ṛgveda and has a very different role from the godhead cult that would develop around the 6th or 5th century BCE. As such, we can probably call this bunk.

2. Like you said, this is pretty out there to begin with and is very weak, if we were to count every figure that kind of looks like a guy sitting in front of another with a bow, we would be drowning in examples.

3. The word "Egypt" comes from Greek Αἴγυπτος or Aigyptos which according to the Oxford English-Greek dictionary comes from Hut-ka-ptah which was old Egyptian for the "House of the Body of Ptah" or "Kingdom of Ptah." The Egyptian name for Egypt was Kemet. So that's one down. Part 2 of this is even more bizarre and an Egyptologist could tackle this better but let me put it this way, Hinduism never spread to Egypt, the Egyptians practiced an eclectic mix of religions throughout their very long history and by the Roman era had become a cosmopolitan place of Hellenistic, Egyptian, Gnostic, Persian, and Christian Religions, with Coptic Orthodox dominating by the End of Roman Rule and continuing to be the dominant, but not ruling, religion until the end of the 1st Millennium CE. This is not to say that the Egyptians were completely unaware of Hinduism, South Indian traders occasionally made there way there, but they didn't stick around.

4. Near as I can tell, there are only Indian sources which claim that it is "ancient" and an "idol" others call it a portable relief. In any case, it is dated to between the 7th and 10th century at a time when there would have been trade with India, so finding an idol or two there would not be especially odd, given that these were precious pieces of art.

5. I'm not sure what to even say about this one other than no. The settlers of America had been more or less cut off from Asia for about 10,000 years by that point and the idea of an "aryan race" is so outdated and steeped in racism that it is pretty offensive. Anthropologists now have completely rejected the idea of race as a biological reality. If we are talking from an ethnolinguistic perspective the answer is still no, because the major point of contact between the Americas and Afro-Eurasia had been severed for well over a thousand years before the PIE language or mythology is known to have existed, let alone what we would think of as "Indian." We can find similar gods just about everywhere if we look hard enough. This process is similar to the psychological process of apophenia, or seeing patterns where there is randomness. In anthropological terms, we are projecting our preconceived notions of what a culture and religion should have onto the culture and religion without considering the inside perspective on it and blocking out the counter evidence.

6. No, I don't know Latin, but I'm pretty confident the answer for this is no, otherwise we would see some echoes of Rāmā or even remotely Hindu style worship there, but we really don't. When Romans encountered Hindu religion, they seemed somewhat confused, but did seem to appreciate it. They seemed most impressed with the Śramaṇas who they encountered since they had a tendency to burn themselves alive, which impressed the Romans. However, they seemed to have had fairly limited contact with Hinduism. This is not to say that there are no links between Hellenistic (and later Roman) philosophy and religion and Indian, far from it, but these connections post-date the historical and mythical founding of Rome.

Now, could Rāmā have existed? Perhaps, the Rāmāyaṇa probably dates back to around the time of the Buddha or just before and was passed down orally for many centuries before being written down. It, unlike some Indian scriptures which are mostly reliable to some extent, was probably modified and improvised by each new teller and saw a huge amount of expansion towards the Gupta era. The oldest versions we have are fairly late, the 10th century CE, but we know it predates that. Insofar as I can tell, Rāmā had a developed cult by the 4th century CE in some fashion, probably as a folk hero, and was incorporated into Vaiṣṇava doctrine sometime after.

I realize that I'm sounding harsh and dismissive here, especially when Rāmā is so dear to Hindus, and I don't mean to. As a historian I'm somewhat obligated to point out baseless claims, especially when they are tied to dangerous ideologies. Rāmā could absolutely have been a real king of a small kingdom during the Mahājanapada era or the late Vedic era. From a historical perspective, he probably didn't fight a massive demon with a king of the monkeys, but that is me speaking as a historian and history and religion don't always mesh well. The only thing I can say for sure is that we have no evidence of him existing outside of the Rāmāyaṇa and that only establishes that an epic was composed, but that doesn't mean that people should just drop everything and say he didn't. I guess I am rambling now, so I'll end this here.

"A vaguely resembles B, ergo B is A," is not a valid argument, it's archaeological pareidolia.

To address even point 5 of this delusion, "similar gods like Hinduism" means nothing. Many deity complexes across cultures have similarities, because the basics of human-environment interactions are similar. Many cultures, for instance, have storm and sun gods, because, as it turns out, many places have exposure to storms and the Sun.

General similarities prove nothing other than the universality of human experience and proximate causes are preferred over more exotic explanations. For instance, the existence of a geoglyph in a region known to have geoglyphs, and the particular glyph appearing to be consistent with iconography of the region, should suggest a local creation for the geoglyph. The hypothesis of the geoglyph being created by an outside entity is an extraordinary claim requiring extraordinary evidence, not vague eyebrow waggling in the direction of several millenia of unrelated iconography.

Likewise, claims of relatedness need to be examined on their evidence, not simply circumstantial pattern finding. It is not enough to state that something begins with "Ram-" and is therefore related to the Ramayana. For one, human speech can only create so many so many sounds and any given system of othrography is even more limited. For instance, the claim that the Inca had a festival called Rama-Sitva relies on an outdated source published in 1884 by someone with no experience in Quechan languages. Even the most generous interpretation of this outdated source would have to point out that there is no such festival called "Rama-Sitva," though there is a festival called Inti Raymi, which is... not linguistically related at all to Rama.

Similarly, any merit of a connection based on "a guy named Senor Vincente Lopez said that there was Aryan race in South America" does not even pass minimal examination. The only reference I could find to such a claim is a citation in the Report of the Commissioner-general for the United States to the International Universal Exposition, Paris, 1900, Volume VI, which appears to be little more than a listing of all the various activities occurring at that particular world's fair. Among such attractions was a talk delivered by one M. V. Henry on the topic of "The Quichua: is it an Aryan Language? A critical examination of the work of Don V. F. Lopez, "The Aryan races in Peru."

Try as I might, I have been unable to locate the text of The Aryan races in Peru, but there is a review and summary of the work in an 1887 work by "an Americanist" named S. F. Walker in the book, The Ruins Revisited, and the World-story Retold. This text cites a review of Lopez by an otherwise unidentified "Mr. Lang" which lays out a highly circumstantial argument for linguistic connections between Quechua and "Aryan Languages." Similar shallow arguments can be found linking American languages to everything from Japanese to Hebrew, and all are similarly without merit. In fact, the text continues by citing a "Dr. Falb" as asserting that "the Quichua and Armara languages 'exhibit the most astounding affinities with the Semitic tongue,'" and that "these Peruvian languages are 'universally Aryan.'" Modern linguistics, of course, place Semitic languages in the Afro-Asiatic language family, which is outside the Indo-European group, but when we are dealing with 19th century blithering about "Preadamite" origins we are not exactly dealing with rigorous linguistic investigation.

Obviously this is just a general refutation on only one of the bullet points you have put forth, but it is consistent with most debunkings of pseudohistory in that the implied connections between cultures evaporate at even the most cursory glance, and that debunking a single sentence requires paragraphs of explanation. As Mark Twain said, "a lie can travel halfway around the world while the truth is still putting its shoes on." Also, Mark Twain never said that, but debunking the origin of that quote would require an entire article unto itself, although the basic truth of the aphorism remains. It is very easy to cherry pick from thousands of years of visual and written works in order to support a pre-ordained conclusion. Less easy is actually learning the history and iconography of a particular culture in order to understand it on its own merits, as opposed to relying on hyper-diffusionist fairy tales.