Sir Walter Elliot is a baronet. In the English nobility, a baronet is usually higher in status than a knight of the realm, but lower than a viscount or a baron. Lady Russell is the widow of a knight, so, in order of precedence, she is lower than Sir Walter who is a baronet in his own right.
Sir Walter's title is not sufficiently powerful to enable him to pass on courtesy titles to his female children (notice the Viscountess Dalrymple's daughter has the title 'Honourable' in front of her name and Elizabeth is 'Miss' Elliot).
Tattersalls, founded by Richard Tattersall in the mid-19th century, was a horse auctioneers at Hyde Park Corner in 1814. Patronized by the upper classes, it was a place for sportsmen to converge, converse, and engage in some gambling.
The establishment has long since relocated, but it is still very much in existence.
Sir Walter Elliot describes the appearance of an admiral he bumped into. As is usual with Sir Walter, the admiral's looks are the primary means by which he judges the man. In contrast to today's standards, tanned skin was then considered unattractive.
Here, in comparing him to mahogany, a wood ages by darkening, he is showing the current interest in things exotic. The mahogany he refers to most likely came from the West Indies, or the Bahamas.
Thus Anne Elliot describes Admiral Croft, the brother-in-law of her erstwhile fiance, to her father, who remains ignorant of most things naval.
Rear admiral is the lowest designation of the highest office in the Royal Navy (admiral). The designation rear admiral of the white means that he is not in the lowest rank of rear admiral, but the middle.
The East Indies, a long way from the West Indies where Captain Wentworth had such luck with the Asp, were made up of the islands of what is now referred to as Southeast Asia, including the Philippines and Singapore.