-She didn't want to use the diamond until it was the very last option. Partly because it was the only thing she had from before she lost her memories, partly because it's a one-time thing - she could presumably keep earning wages, but once the diamond was sold, the money would have been finite, and then once she was in Paris, she would have less guarantee of work.
-Russian aristocracy was WEIRD when it came to affairs. They often tended to look the other way; it was even quite common among some of the more bohemian nobility. But Lily wasn't a direct Romanov (she was Maria's cousin, and Maria wasn't a Romanov, she married into it), and she was just one of many nobles/royals at the time. Anastasia is the direct descendant of the royal line, as well as the last true Romanov left and the family's heir. She would probably be pressured to marry a blue blood to maintain social class, as well as to avoid the appearance of an impoverished suitor only seeking to marry her for her money. A cute but poor romantic partner would be a "luxury" no longer available.
-She comes in without a coat/shawl/stole, so maybe she stopped in the coat room first, whereas the men didn't or got done there faster? I think that's how the animated movie framed it.