Edit: I know that Leif Erikson founded some small settlements in what is now Canada, but as far as I know, he didn't create any permanent colonies there. Correct me if I'm wrong.
It's very difficult to find civil discussions about the matter. For example, this link that I found started out with some decent arguments from both sides, but then insults started being thrown around, and it eventually the topic changed to Asian Indians.
The thread I linked to has a few pictures of bearded statues, but the guy who posted the pictures of the bearded statues didn't provide sources, he just said that statues were Mayan.
Besides claims that ancient American bearded statues don't look like Native Americans, another theory the Euro-centrists like to bring up is the Solutrean hypothesis, which claims that European Solutreans were the donor culture of the Clovis culture. I really don't know much beyond that. The Solutrean hypothesis doesn't seem to have much evidence backing it up.
Arguments about bearded statues tend to revolve around whether or not Native Americans are capable of growing beards. Also, people rarely post sources of the bearded statues they post. And often times, doing backwards image searches leads me to blogs, which also don't provide sources. I know of a few examples of American archeological sites with bearded statues, but those are just the ones I found on my own. Does anyone here know anything about the bearded statues that were posted in the thread I linked to? Where were they found?
What subreddit would be good for asking which pure-blooded Native American ethnicities can grow beards? Every time someone posts a picture of a Native American with a beard, someone else will claim that the man is mixed race. Is there anyway I could find the DNA test results of a bearded Native North/South American man?
Feel free to recommend subreddits where you think this topic is relevant.
You don't have to focus your arguments around bearded statues and the Solutrean hypothesis. I just mentioned those because those are the two main things that Euro-centrists bring up when discussing this topic.
Imagery of bearded individuals is not uncommon pre-Columbian art. There are plenty of examples from Mesoamerica and the Andes, but I'll let our local experts from those areas discuss specific examples. Instead, I'll tackle the misconception that Native Americans can't grow facial hair.
During the colonial era, facial hair was largely out of fashion for a variety of reasons. Among the Powhatans, for example, there were laws that prohibited most men from growing out their beards. Only certain priests and older men were granted that privilege, and Powhatan himself had a beard. Among the Iroquois, a common stereotype cast men with facial hair as uncouth and dim-witted, so keeping your face smooth was important for cultivating an air of civility. Ethnographic sources are full of descriptions of Native men industriously keeping their facial hair at bay with constant plucking, usually with shell tweezers. I'm away from my sources at the moment, so in the meantime this older question will help shed some additional light on the issue.
As for the Solutreans, I've dealt with them and other popular pre-Nordic visitors from across the Atlantic in other posts: the Welsh, the Israelites, the Minoans, and the Solutreans.
If you have any more specific questions, I can try to deal with those a little later.
None.
/u/Reedstilt has already linked to several past discussions debunking pseudohistorical/anthropological hypotheses about any number of Eurasian groups in the Americas, so I'll just stick to making a couple of specific points.
First, Native Americans (and East Asians/Siberians) are fully capable of growing facial hair, although it does tend to be relatively less dense. Our most famous example of this is actually Motecuhzoma Xocoyotl. Bernal Diaz del Castillo (trans. Carrasco 2008) described him as having a thin, neatly trimmed beard.
Second, that forum post is ridiculous. Aside from the entire "anthropology" section apparently being based on 19th Century racist nonsense, there's no need for bizarre theories about "albino Amerindians" to explain red-haired mummies. Post-mortem lightening of hair color is a well-understood phenomenon.
This information comes from Scientific American Frontiers on PBS. Feel free to watch online. The hypothesis has one very convincing piece of evidence, that for me is the smoking gun. If native Americans crossing the land bridge from Asia had developed arrow heads and spear heads, there would be a trail of artifacts from the Pacific Northwest all the way back to Russia. But there isn't. The arrow heads and spear heads only begin to emerge in the desert Southwest. And the similarities between the Solutrean's and those found in the Southwest are overwhelming. It's not racist. It is a plausible theory that should get more attention and research.