The short answer is that the listed options are very interchangeable indeed. The choice of one over the other primarily has to do with the personality of the person talking. I will try and help you understand the nuances of the options you have provided, but I want to be clear that the differences between them are so small that other folks may not agree with my assessments.
Living abroad for years helped him have a host of experiences.
"A host of" originally/technically refers to a group of people. While it's completely normal to talk about a group of experiences this way, one's mind almost personifies them a little bit. Like each experience has a little something unique to add, though there are many of them.
Living abroad for years helped him have an array of experiences.
An array of something is usually a number of items that can be set out in an orderly fashion. There's a sense of spreading these experiences out in front of you, comprehending them all and perhaps making sense of them. Though there are many of them, they can be categorized and perhaps counted.
Living abroad for years helped him have a myriad of experiences.
Myriad usually means so many you can't count or keep track of them. It's an overwhelming number. The nuance here is different from "an array" above. You could also say "myriad experiences" without the of. By the way, myriad is not that common in spoken/informal English. Uneducated people might think you are being snobby if you use this word.
Living abroad for years helped him have a multitude of experiences.
By saying there is a multitude of experiences, I'd say we are focusing on how different the experiences are from each other and from one's experiences from one's home country.
Living abroad for years helped him have scores of experiences.
A score is 20. If you have scores of something, you have handfuls and handfuls. The focus here is on how many you have, though perhaps not as overwhelming a number as myriad.
