While this topic is a little before my expertise, I can at least attempt to answer some of your questions. To make my answer shorter, I will simply refer to Anglo-Saxons as the English.

1) What was the initial reaction of the ordinary anglo-saxon?

I don't know of any sources that have a record of a peasant's opinion, but we can make an educated guess from some information. We know that warfare was a constant between the Danes and the English. Here are some (of many) excerpts from the Anglo-Saxon Chronicle (ASC) demonstrating just how often they fought.

“A.D. 833. This year fought King Egbert with thirty-five pirates at Charmouth, where a great slaughter was made, and the Danes remained masters of the field. . . . A.D. 840. This year King Ethelwulf fought at Charmouth with thirty-five ship's-crews, and the Danes remained masters of the place. . . . A.D. 902. This year was the great fight at the Holme (39) between the men of Kent and the Danes. . . . A.D. 943. This year Anlaf stormed Tamworth; and much slaughter was made on either hand; but the Danes had the victory, and led away with them much plunder.” (The Anglo-Saxon Chronicle)

There is a lot in the ASC about the Danes and English fighting, both on land and on sea. We know from this that the “ordinary anglo-saxon” would have been occupied in this constant state of warfare. We can make an assumption that there was likely not a generally friendly attitude towards the Danes.

To further cement this idea, that the English populous were probably not too friendly with the Danes, I’ll turn to Marc Morris, author of ‘The Norman Conquest.’ He writes, “The society [circa. 1000 CE] - slaves, ceorls [peasants], thegns [lesser nobles] and ealdormen [higher nobles] - had been severely shaken by the Danish invasions . . . naturally, the population as a whole had suffered as Viking armies hacked their way across the landscape.” (Morris, The Norman Conquest, p. 27) This illustrates that all levels of society were affected negatively by the Danes up into the 11th century.

2) Was the relationship between the Anglo-Saxons and Danes in Danelaw and the rest of England bitter, or did they live together peacefully?

I feel like this question may already have been given above, but I can add a bit more. During the Danelaw, in areas ruled by the Danish, taxes would be collected. Reading Thomas Wright’s book, ‘The Celt, the Roman, and the Saxon,” I have found that there were some more civil-type disputes between the Danes and English. Specifically, in the payment of taxes. Wright explains that many English towns had formed after the conversions of many Saxons to Christianity. These towns may have received their rights and protections more closely from the church than a lord.

This proves problematic for the controlling Danes who don’t care about the town’s perceived history and rights. Wright writes, “When the Danish king Swegen, that at Gainsborough, ignorant or careless of local privileges, demanded a tax of the people of Bedricsworth, the [people] pleaded their exemption from royal taxes, and refused to pay.” (Wright, The Celt, the Roman, and the Saxon, p. 447) Later, the ecclesiastical leaders of the town collected taxes from the people, because in the minds of those English people that was who was rightfully supposed to collect taxes from them.

Hopefully, this helped answer your question. I would say that the average Anglo-Saxon probably did not like the Danes due to constant warfare, being the victims of conquest, and a trampling (within the Danelaw) of their perceived rights.