It's quite difficult to give a complete account of the Laki eruption and its consequences in a post as short as this one since the questions posed in the opening post might well serve as the basis of a PhD thesis. The eruption had far-reaching consequences and its effects were felt not only in Iceland and Northern Europe but also in areas as far away as the United States and Egypt. The eruption is believed to have caused a global drop in temperatures and some scholars have even suggested that it might have been a contributing factor in the French Revolution (due to the resulting crop failures in France). It led to the death of around 20-25% of the Icelandic population as well as the death of around 80% of sheep in Iceland, 60% of horses and 50% of cattle. The effects were cataclysmic and the resulting hardships are known in Iceland as 'Móðuharðindin'. Scholars do not agree on a translation for the term but a few of the suggested ones are 'The Mist Hardships', 'The Blight of the Blur' and 'The Famine of the Fumes'.

Were there any populated areas that got covered over by lava at the time?

Although the eruption took place quite far away from inhabited areas, the lava output was huge (around 14km^3) and it covered a gigantic area of almost 580km^2. It was followed by an expulsion of tephra, acid rain and toxic fumes which created a 'blur' or 'mist' which hung heavy over Iceland and poisoned both people and livestock. These toxic fumes would later spread to Europe where they caused crop failures and drought in the Northern part of the continent. Many of the people who lived near the volcano in southern Iceland were forced to flee the eruption. The lava completely destroyed 13 farmsteads and much of the arable land in the area. Almost 47 estates in total were abandoned due to fear of the lava and inhabitants were forced to seek refuge elsewhere in Iceland. Their farms would remain abandoned for the next couple of years and 5 years after the eruption had ended, many of the farms had still not been resettled.

What did people in Iceland think of the eruption?

Volcanic eruptions in Iceland are quite common and many of the people living in Iceland at the time would have been aware of the destruction that volcanoes were capable of causing. Mt. Hekla had erupted in 1766 and Mt. Katla had erupted in 1755. Both eruptions had caused a considerable amount of damage though neither of them rivalled Laki in terms of destruction and death. People would have been afraid of the dangers that the lava flow presented and the effects that it might have on their livestock and land. The toxic fumes would have caused a burning sensation in their throat/nose and for people who worked outside, breathing would have been difficult and downright painful. For a look into the minds of people who experienced the eruption, it's worth mentioning the account of Reverend Jón Steingrímsson, who was a pastor in the nearby town of Kirkjubæjarklaustur. He described the eruption in a journal entry made a few weeks after the volcanic eruption began:

"The 20th of July 1783, which was the 5th Sunday after Trinity Sunday, was marked by the prevalence of haze, thunder, lightning and earthquakes. But because the weather was calm, I and everyone else who was in the area of Síða, inhabitants as well as visitors, decided to attend Sunday service, despite fears and mournful thoughts that it might be the last service ever held in our church due to the threat that had arisen and drew ever closer, and looked like it might destroy the building like so many others before it. When we arrived at the church, the haze and smog that arose from the lava now occupying the riverbed, was so thick that we could not see the church from the monastery doors, unless it was temporarily lit up by lightning. The thunder and lightning were so great and in such abundance that they echoed inside the church and rang the church bells and the earths tremors were relentless."

(Own translation)

When churchgoers left their church after service, they noticed that the lava flow had spared the building by coming to a standstill just a few kilometers away. They believed that it had been an act of God and that Reverend Steingrímsson had stopped the lava flow with his 'fiery sermon'. After this event, he became known as 'eldklerkur' or 'Fire Cleric'. Though the church might not have been spared due to divine intervention, it shows just how frightened people were of the eruption and the lava flow which threatened to destroy their farms.

Did any word from Iceland come to mainland Europe about what was happening, and if so what were other Europeans’ reactions to hearing about it?

News of the eruption's devastating impact arrived in Denmark at the end of August (the eruption had begun in early June). Danish officials wanted to help the beleaguered Icelandic population and arranged for a ship to sail to Iceland to gather information on the full extent of the eruption. Danish official Hans Christoph von Levetzow, Bailiff of Iceland, and Icelandic student Magnús Stephensen sailed from Copenhagen on October 10th but were unable to reach Iceland before winter began. They were forced to wait until spring in Norway and did not arrive in Iceland until 16 April 1784. The information that the Danish authorities had received in late 1783 was, of course, not very extensive and because of that they failed to realize the full extent of the situation in Iceland. They were not aware of the impending famine and did not realize that urgent help and food assistance were needed. When they finally realized the direness of the situation, they donated both food and money to the areas hit hardest by the eruption. The Danish public also helped by donating 46,000 rigsdaler which was a considerable amount of money at that time (for some context; a cow was worth roughly 7 rigsdaler). Despite these efforts, however, thousands of Icelanders perished as a result of the eruption. The Danish authorities did what they could but due to poor transportation and communication between Denmark and Iceland, the news of the disaster did not travel fast enough.

In the aftermath of the eruption, some Icelandic officials, such as Jón Sveinsson (a magistrate in a nearby county) suggested relocating parts of the Icelandic population to Jutland in Denmark. Bishop Hannes Finnsson later claimed that this proposal had been seriously discussed in 1784, but ultimately rejected by Danish authorities. The Danes did, however, suggest in 1785 that infirm and old people (a tally of 500 was mentioned) might be transported to Denmark to live out their days so they would not keep on being a burden on Iceland, but this plan was rejected by Icelandic officials.

Sources:

Dansk biografisk leksikon X, edited by Povl Engelstoft and Svend Dahl. Copenhagen: Gyldendal, 1982.

Saga Íslands VIII, edited by Sigurður Líndal. Reykjavík: Hið íslenzka bókmenntafélag, 2006.

Safn til sögu Íslands og íslenzkra bókmenta að fornu og nýju IV. Reykjavík: Hið íslenska bókmenntafélag, 1915.