I can't help you with the managing kids at your elderly relatives bit but I can encourage you with the flight part. It's absolutely doable with kids that age, and you'll have so much fun once there, it will be well worth it.
Flying is really not a big deal if you prepare yourself right and set expectations. I have flown with my two DC many times, starting when they were just babies/toddlers, about the age of yours. I was the only adult traveling too, with two toddlers/young kids on international flights with layovers and, honestly, it was fine. Not super ideal, obviously but, then, I don't particularly enjoy 12h+ international flights in economy on my own either, it just is what it is and you get on with it, and then it's done and you enjoy the destination.
I've flown a few different international carriers and most all were really kind to the kids (ok, SAS might have been my least favorite experience, but, man, were they cheap!). Many had little activity packs they gave out to keep the kids occupied. With every flight I knew more what to pack so things went smoother too. When they were little, I would have a new little trinket or toy every hour or so, snacks packed in small boxes, later iPads with headphones etc. I did always buy the younger one a seat, because traveling alone with a toddler AND a lap infant is hard, much easier when they are in their own safe and secure car seats. I really wanted to go the places we were going to, and it was the three of us now, and economy was all we could afford, so we made do. I told myself economy was the only place that always had 3 seats in a row anyway and just bucked up :). Ultimately, economy, business or first class, it still comes down to the fact that you're going to be confined to a relatively small space for about half a day and it's not super great but the reward is worth it. So as long you as you get the living situation squared away, don't even think twice about the flight part, you can absolutely do it, the flight will be over before you even know it!