First, it's important to distinguish between the Clovis culture and "Clovis people". The Clovis culture is a well-defined archeological entity, consisting of characteristic features such as the bifacial Clovis points and a big game hunting lifestyle. The "Clovis people" is a more amorphous and less well-defined thing.

Archeological evidence of the Clovis culture is found from the period of around 13,000 - 12,500 years ago, in the form of Clovis points and butchered remains of animals they hunted. However, humans remains associated with the Clovis culture are rare. The only human remains incontrovertibly associated with the Clovis culture that have been found belong to Anzick 1, the skeleton of a boy found in western Montana. Anzick 1 dates to about 12.650 years ago, and genetic tests show that he was more closely related to modern native Americans than to any other existing population.

So, getting back to your question:

1. "Clovis First" in the sense of the Clovis culture is clearly not true, because we have found archeological evidence of humans in the Americas that predate the Clovis culture. Genetic evidence shows that humans first entered the Americas long before the Clovis culture.

2. But what about the Clovis people? Were the people who eventually invented the Clovis culture the first inhabitants of the Americas? This question is more iffy, for two reasons. The first is that we don't have a large sample of "Clovis people" - in fact, we have only one skeleton that is known to have belonged to the Clovis culture. If this one boy was representative of the Clovis people, then yes, his ancestors could well be among the first inhabitants of the Americas, based on genetic evidence. However, this does not mean they were the only members of that "first inhabitants" category.

Generally, the problem is that there is very little archeological evidence of the earliest people in the Americas. There are a bunch of sites in both north and south America that some scientists claim are very old, but their dating remains controversial. So we are forced to rely unduly on genetic evidence, which is only approximate, and sometimes unclear.

There are a couple recent papers (from last month) that cast some light on this subject:

1. Genetic evidence for two founding populations of the Americas

2. Genomic evidence for the Pleistocene and recent population history of Native Americans

They agree on some points but differ on others. Both agree that native American populations diverged from their cousins in Eurasia around 23,000 years ago, living in a relatively isolated region known as Berengia. This was the land bridge between Asia and North America, where the Bering Strait exists now. At the time (during the last Glacial Maximum), sea levels were much lower, and a large land mass existed in this region, which was home to these Berengian people.

Both agree that there were no more than 8,000 years between the divergence of the Berengian people and the peopling of the Americas. So the earliest humans arrived in the Americas no less than 15,000 years ago. How much earlier is unknown, but going by these numbers we can say 15,000 - 23,000 years before the present are the two extreme limits.

Both agree that almost all native people in the Americas are descended from the first major wave of migration, somewhere in the order of 80% of native Americans alive today. Probably there were a couple successive waves after the first in the distant past, as well as more modern waves (such as that which brought the Inuit people).

So we can say that the bulk of native American ancestry belongs to these early waves of migration from Berengia, between 23,000 and 15,000 years ago. This includes "Clovis people", as represented by Anzick 1. Of course, they did not immediately develop the Clovis culture, which came much later.

One important way in which the papers diverge is that the first paper finds evidence of a second population group, that is more closely related to some Australonesians than to other native Americans. The modern representatives of australonesians include the aboriginal population of Australia and New Guinea, and the Onge people of the Andaman Islands. The people related to them are now found in a few places in the Amazon rain forest of Brazil, so they are a tiny minority among the native people of the Americas. However, the paper says that this ancestry is equally old -- meaning, the Australonesian admixture is not modern, it doesn't come from Pacific Islanders making trips to the Americas in historical times. They say that the best explanation is that the people of Berengia consisted of two subpopulations, one more closely related to modern native Americans, and the other more closely related to Australonesians. They both migrated into North America from Berengia, 15,000+ years ago.

More research is needed to pinpoint when this Australonesian population in Berengia diverged from their Australonesian ancestors who populated Australia, New Guinea or the Andamans.

Note that this is all genetic evidence, which comes with a lot of caveats. The dates are approximate. It's based only on native Americans alive today plus whatever archeo-DNA we have been able to find among human remains in the Americas.

So in summary: Clovis First as referring to the Clovis culture is not accepted anymore. Clovis First referring to the Clovis people is somewhat true, so long as we acknowledge that they weren't the only ones, there is some Australonesian ancestry there too.

As for your question about the Solutrean Hypothesis, no it hasn't gained any currency among archeologists or anthropologists. It was always a fringe hypothesis, with little to no support except for a resemblance between Clovis and Solutrean points. It has never been mainstream, and modern genetic evidence keeps making it more and more implausible as time goes by.

If you take a stroll through the interwebs you'll find that the "Solutrean Hypothesis" is mainly only accepted by White Nationalists and others of their ilk. It's based in their fervent fantasy that white folks got everywhere and did everything before anyone of darker skin could have managed it. This article gives a good summation of what the hypothesis is and why it's really an impossibility.

It might be worth crossposting this to /r/AskAnthropology.

Here is a thread from /r/AskAnthropology that discusses pre-Clovis sites and timing of early New World settlement. I don't think there is much persuasive evidence for the Clovis-first argument. Nor do I think it has much of a following among archaeologists. I would look to David Meltzer's First People's in a New World, 2010 for a thorough discussion.

As to the Solutrean Hypothesis, Meltzer observes:

"Few if any archaeologists—or, for that matter, geneticists, linguists, or physical anthropologists—take seriously the idea of a Solutrean colonization of America"

I'd be interested to know as well. The PBS special First Peoples seemed to suggest that the most likely scenario was a concurrent migration via both land and sea from Siberia. I had never heard this combination of theories before. Could anyone speak to that?