Hi there. To obtain an excellent credit rating you need to have a score between 961 and 999 points. Don't obsess over the score though, it is Experians own rating purely for your own reference. When you are actually credit checked lenders will use the report information plus any other details you've given them (age, employment details, salary, marital status etc) to come up with their own score.
It seems that these missed payments with Egg haven't affected your score that much so they can't be that severe. There are two things you can do to dispute the status history of the settled Egg account. Firstly, contact Egg to see what they are willing to do for you (if you don't know the card number, contact Experian and they will provide it for you). If Egg are unhelpful or you don't want to contact them directly, dispute the information through Experian who will raise a query on your behalf. There's no guarantee that Egg will respond to Experian with a result that is in your favour so I suggest contacting them first.
As for the insurance searches you'll be glad to know that they are unseen by every company who searches your report other than Royal and Sun Alliance themselves. This is because the search type is an "unrecorded enquiry" and these search types are for non lending purposes ie insurance quotes, account management etc. They do not affect your score either and it's actually standard practice for insurance companies to search your report, look at your finances and then base a quote on how you manage your accounts. Insurance searches will appear on your report as either Royal and Sun Alliance or Admiral Bell Diamond Elephant.
Hope this helps.