Your specific questions definitely deserve expert input, and while you are waiting for them to craft their answers you may want to take a look through some previous threads on the Vikings in North America:

Viking Colonization

• Do we know now that Vikings landed in America? - 44 comments, over 12 months old.
• In this thread, a flaired expert addresses a variety of myths surrounding the extant of Viking expansion in the Americas.
• What stopped the viking colonization of america? - 12 comments, over 5 months old.
• /u/textandtrowel in a comment further down in the thread gives a survey of the archaeological evidence present at L'Anse aux Meadeux which you may find interesting.
• Why didn't the Vikings establish a permanent colony on Greenland or North America? - 31 comments, over 1 year old.
• The top level comments discuss the North American colonies in context with the Greenland settlements.
• What did Vikings eat while on their voyages to the Americas? - 7 comments, over 10 months old.
• This is rather tangential to what you are asking, but one flaired user gave a very decent answer to it that makes for a good read.
• How far into North America were the Vikings able to travel? Could they have made it to the Yucatan and had contact with the Maya? - 10 comments, over 1 year old.
• The commenters here put the (lack of) evidence in context with the Native American records.

The Vikings and the Native Americans

• Do any Native American/First Nations peoples have any oral traditions or tales about the Vikings? - 118 comments, over 1 year old.
• One user in this thread gives summaries of the Kalaallit (Inuit people native to southwestern Greenland) oral histories related to the Vikings.
• When the Vikings settled in Greenland & Vinland, they encountered the ancestors of the Inuit and more than once had skirmishes. Why did the Natives emerge victorious? - 7 comments, over 1 year old.
• In this thread commenters talk about the motives that that the Vinland settlers had and the many challenges that they faced in North America despite the marginal advantages of their iron weapons.
• Did the Vikings spread diseases to the Native Americans when they first explored North America? - 10 comments, over 1 year old.
• The users here put the Vikings in context with European diseases and the population density of Newfoundland to show why this devastation did not occur at that time.

The Spread of Tales of the Vikings' Travels

• Did word of the New World discovered by Leif Ericson ever get to the Western Europeans? - 65 comments, over 1 year old.
• Here the commenters pull out sources from Germany and other areas to show the state of awareness in Europe of the New World before 1492.
• Did the rest of Europe know about the Vikings' discovery of America? - 8 comments, over 1 year old.
• This thread puts the above thread in context by summarizing how Scandinavian scholars did not realize that Vinland and the areas that Columbus explored were one in the same until the 16th century.
• How did the Nordic countries react to the 1490-1500s New World discoveries? - 32 comments, over 1 year old.
• Users his talk about the internal politics and goals of the Scandinvian countries contemporaneous with Columbus's discovery and why they focused on nearer shores.
• Is there any evidence that Christopher Columbus (and other 15th century explorers) knew of Lief Erickson's travels and the Viking settlements in the New World? - 4 comments, over 1 year old.
• A flaired user gives a well sourced answer that explores Columbus's mindset and preconceptions that would have shaped his interpretation of the Vinland voyages.

The Vikings did travel to North America after their discovery of Greenland (to which had their own native peoples). A viking ship had been sent off course in the 10th century. First area of discovery was called Halluland (Slav Land). Moving further south, there was the Markland (containing forestry and woods).

According to Viking Sagas - the written history of the Icelandic Viking people who developed their own written language - Leif Erikson had visited North America. This would be the earliest account of a European (albeit not called Europeans at the time) visiting North America.

Historians believe that Vinland was modern day Newfoundland. There was evidence to suggest that L' Anse Aux Meadows was a Viking settlement.

In terms of the argument for the waters, there was a warming period in which the vikings set off for North America. This not only allowed for the settlements on Greenland (which was covered in ice) but also to launch off of it.

For the last two questions - History is almost, in my belief, purely subjective in these instances. It is GENERALLY accepted by historians that Vikings were the first to travel to North America. Counterarguments do exist, but the general consensus is that the Vikings did indeed perform this travel.