Learn English – Does the “elbow-handshake” have any relation to the phrase “rubbing elbows”

According to World Wide Words, it originates from firemen doing speed competitions without carrying water.

The term run, more fully fire run, has for at least the past century been used by local fire departments in the USA for a call-out to the site of a fire. It was once common for fire departments or volunteer hose companies to give exhibitions of their prowess at carnivals or similar events. [...] These competitions had fairly standard rules, of which several examples appear in the press of this period, such as in the Olean Democrat of 2 August 1888: “Not less than fifteen or more than seventeen men to each company. Dry run, standing start, each team to be allowed one trial; cart to carry 350 feet of hose in 50 foot lengths ...”.

These reports show that a dry run in the jargon of the fire service at this period was one that didn’t involve the use of water, as opposed to a wet run that did. In some competitions there was a specific class for the latter, one of which was reported in the Salem Daily News for 6 July 1896: “The wet run was made by the Fulton hook and ladder company and the Deluge hose company. The run was made east in Main street to Fawcett’s store where the ladders were raised to the top of the building. The hose company attached [its] hose to a fire plug and ascending the ladder gave a fine exhibition.”

It’s clear that the idea of a dry run being a rehearsal would very readily follow from the jargon usage, though it first appears in print only much later.

-ship is added to a noun to establish status or condition.

1. Indicating a state or a condition, e.g. to be in a friendship.
2. Indicating the qualities belonging to a class of people, e.g. craftsmanship.
3. Indicating office or profession, e.g. ambassadorship.

Quote from http://www.randomhouse.com/wotd/index.pperl?date=19990225:

The suffix -ship has been studied in some detail, but most of that detail is irrelevant to all but the most hardcore etymologist. The most important thing, in terms of its origin, is that it is unrelated to the word ship meaning 'a vessel, especially a large seagoing vessel'.

-ship is a suffix used to form nouns of state or condition, chiefly added to nouns and especially personal nouns. In Old English it was widely used with adjectives and participles, but only two of these survive (hardship and worship, from an adjective meaning 'worthy').

The uses can be divided into ever-finer distinctions, but there are some basic categories. It can denote 'quality; condition' (kinship; friendship 'the condition of being a relative/friend'). It can denote 'skill; act; power' (scholarship 'the acts of a scholar'; horsemanship 'the skill of a horseman'). It can denote 'number' (readership; listenership 'number of readers/listeners'). It can denote 'profession; office; position' (professorship 'the profession of a professor'). It can denote rank or title (ladyship; lordship).

The suffix was common in Old English, and is ultimately a form of the ancestor of the Modern English verb shape, in the sense 'to create; form'. Cognate suffixes are found in many Germanic languages; a form occasionally encountered is the German Wissenschaft 'science', from Wissen 'knowledge' and -schaft '-ship'.