I don't have any problems with my 243 H&Rs, I have 4 of them, two Superlights and two Ultras, one of which I had rechambered to 6mmAI, all shoot excellent, ejector barrels have a bad reputation in general for high pressure rimless chamberings, but even that can be dealt with 3 of mine are ejectors, the only time I've had a case stick was my fault for not wiping the chamber after cleaning at the range. All recent production barrels are extractors making sticking brass a moot issue. The 243s do tend to show a preference for lighter or heavier bullets tho, some like the heavier 95-100gr loads, others seem to like them 90gr or lighter, but that can be said for a lot of rifles, just depends on the barrel.
There are some great 223 deer bullets that have been used very successfully, the 53gr Barnes, 60gr Partition and 64gr Win PP for a few, hopefully some of the fellas that use them will chime in, the 223 isn't legal for deer here, so I can't speak personally on it, but there have been many discussions on it here, the success of others using it have changed my view on it, it's recommended by many state wildlife depts for their youth hunts, with good bullets and proper shot placement and range, it's ideal for little people.
410 Slugs aren't very highly respected for deer if you read in the H&R shotgun forum, but I have no use for a 410 to begin with, so I have no experience with it, so take it for what it's worth, just what I've read there.
Tim