"The action of obeying, such as bowing or curtsying, to show respect" (OED).

"The action of diminishing or making less; the process of diminishing or becoming less" (OED).

Similar in style to the modern day convertible, the barouche-box was a four-wheeled carriage with a falling top. It had two sets of double seats, positioned to face each other, and a seat for the driver, called the box, outside of the carriage. Due to its light, somewhat flimsy design, it was regarded as a summer carriage.

In Austen's time, the importance of passing down a legacy to a male heir is heavily stressed. Oftentimes things like property and money are passed down to the nearest male heir, even if that means skipping any direct children for inheritance. It is very rare for a woman to inherit.

Curtsying is a sign of respect typically used by those of lower status to address those of higher status. In this case, Anne De Bourgh is showing deference to Elizabeth and Lady Catherine curtsying first, which is somewhat strange, considering she is of higher status. By holding her hands out, she is also expecting a similar show of respect back.