Funnily enough, I came across an answer (of sorts) to this second question just yesterday when looking something up for another thread. According to Donald Keane, no one is quite sure why infant mortality was so bad, but the factors most often cited are:

1. Future emperors married extremely young

2. Court physicians were quacks

3. The court had an "unhealthy, gloomy atmosphere"

to which Keane adds another possibility which "is rarely suggested," although you nailed it in your question:

4. Very few noble women of adequate rank -> inbreeding

I might add that Meiji's father Komei actually died at 35, although it was from smallpox which seems to have been, you know, going around at the time.

For more, check out Emperor of Japan: Meiji and His World, 1852-1912