How difficult would it be to study English if my native language were from a completely different language family (Non-Indo-European)? Such languages have completely different grammar structures, lexical roots etc. Can this actually be a plus?

Every language is hard to learn for non-native speakers. Generally speaking, languages more closely related to your native language will be easier to learn. For example, a Spanish speaker would have an easier time learning Italian than learning Mandarin Chinese. All languages pose a challenge, but English is no different from other languages and likely isn't much more difficult than other languages related to English such as German, Dutch, and Swedish. English's odd spelling system may make English harder to read, but the language itself is no more or less complex than any other language. All the language have their different grammar structures and complexity I don't think that really works, English have there own terms of grammar if you want to be a good writer you must follow them strictly.