Furthermore, could a literate Norseman in England read Anglo-Saxon runes?
Anglo-Saxon Runes are a slightly odd beast when it comes to their prevalence, because they were never particularly common in comparison to the runic inscriptions that are found routinely in Scandinavia. Inscriptions in the Anglo-Saxon Runic alphabet, also called, fuþorc, for its first five letters, are not common in comparison to the rich runic inscriptions of futhark runes, which were used in Scandinavia and some continental settings through to the High Middle Ages (though modified as time went on and the languages of Scandinavia further differentiated before becoming the Medieval runic alphabet). Elder Futhark inscriptions are admittedly significantly less common than Younger or Medieval futhark.
In contrast to the widespread use of runes in Scandinavia, the runes of the Anglo-Saxon inscriptions are much more constrained in their use. They were most common in coastal communities, and cluster on the Eastern coast of England with some limited usage in areas like Kent in Southern England (not far from the coast either), however they were used widely when it comes to the things that they were carved onto.
Among the surviving corpus of Anglo-Saxon runic inscriptions (a hundred and change or so when all is said and done) are found items as diverse as weapon paraphernalia, jewelry, antlers, coins, kitchen utensils, and stone crosses, so there does not seem to have been a particular restriction of these inscriptions to certain items, nor were the limited by the adoption of Christianity among the early English peoples. Items such as the Frank's Casket contained runic inscriptions alongside scenes that contain allusions to Germanic myths, such as Weyland the Smith, as well as to Christian stories such as the adoration of the Magi (the Three Wise Men of Nativity fame).
The usage of the runic alphabet by the Anglo-Saxons was, as mentioned, not common, and following the introduction of the Latin alphabet the trend was over time for the adoption of the Latin letters (with three exceptions which were retained forthe sounds they made in Old English, these were the letters þ, ð, and ƿ). However runes never really fully disappeared in Old English writings, with the last use of runes in Old English coming in a 12th century manuscript that post-dates the conquest of England by the Normans. However, by this point runic inscriptions were rare, the bulk of Old English runic inscriptions date to before or the early parts of the Viking Age, roughly the 600's to the 800's.
Now as for the mutual intelligibility of Old English runes with Old Norse, that is really two questions. Could a hypothetical literate viking read Anglo-Saxon runes, and could he understand what they said. These are two different questions as familiarity with an alphabet does not inherently imply understanding of what is actually being written. (For example I can read Greek letters, but I cannot read Greek beyond a few certain words and endings much to the disappointment of my undergrad Greek professor). The runic alphabets of Scandinavia and England in the Viking Age were not necessarily mutually intelligible. Younger Furthark contained only 16 letters, while Fuþorc contains 28ish. (Not all runic alphabets from the period use the exact same runes for the same sounds) Many letters that had been elided into one rune in Younger Futhark remained separate in Fuþorc, which presents obvious problems for mutual intelligibility if almost half of the alphabet is different.
On top of that is the issue of the mutual intelligibility of Old Norse with Old English. Although this is more likely than you may thing, Old English, especially in the northern parts of England and along the coast was heavily influenced by Old Norse (for example one of our basic words in English "she" comes from Old Norse) but the actual mutual intelligibility of the two is uncertain. Its possible that when both were spoken widely that there was more mutual intelligibility because of exposure. However just going off of written versions of the language, they are not very mutually intelligible.