Eggs are really hard to see, especially when looking through a veil. Instead, concentrate on finding larvae of various stages. Newly hatched larvae will be in a pool of blue-ish white royal jelly. The older larvae will be quite obvious. With only the one hive, you really need to find a local mentor that can help you with a thorough hive inspection. It is not necessary to go deeply into the hive very often, but a quick check for food stores and brood should be done no less than once per month IMO. If something does not look right, a deeper inspection is called for. Even on the quick inspections, check both brood boxes. I often find the top box packed but very little going on "downstairs". Good time to rotate the boxes. The good news for you is that you appear to have a fair amount of capped and emerging brood, so the queen is alive and laying. But, you need someone local to help you evaluate the hive's overall condition.