Learn English – Is double prime (″) the correct symbol to use for specifying inches

How can I pronounce @ symbol: At / At the rate?

The "at mark", "at sign", or "at symbol" is its usual name. According to Wikipedia its official name is "commercial at".

Can I use it in a sentence? Please explain with an example.

It has no function in English sentences.

It seems the best current scholarly theory is "none of the above." The theories cited in the question have been proposed by various scholars, but there is little evidence to back any of them up.

However, another proposal came out of the work of Italian scholar Giorgio Stabile in the late 1990s. (The story is basically told here. Unfortunately, while there are a number of corroborating media stories carrying this information, I couldn't find a more scholarly publication, though there are a few interviews or short pieces by Stabile on the internet mostly in Italian.)

Stabile started by tracing the names of the symbol in various languages and discovered that many seem to use terms that just describe the shape in some colorful way, employing words that translate as things like "elephant's trunk" or "pig's tail" or "worm" or "rose."

But there was one pattern of names: French arobase, Spanish and Portuguese arroba, and Italian anfora, which are all archaic terms for old measurement units. (These are more formal names; French and Italian also use informal terms like escargot and chiocciola for the symbol, both meaning "snail.") The Italian word is related to the Latin amphora, which was both a name for the storage jars and a standard unit of measure. The French, Spanish, and Portuguese words ultimately come through Arabic from al rub', meaning "one fourth" (of a hundredweight). All of these words referred to general measurement units in the Renaissance.

The final link for Stabile came with a discovery of a 1536 letter sent from Seville to Rome by a merchant Francesco Lapi, in which Lapi discussed the price for amphoras of wine. In his letter, Lapi used the @ symbol as an abbreviation for amphora (or anfora). (A scan of the document can be found here.)

The swooping curve around the "a" fits in well with similar curved gestures in Italian handwriting and letterforms of the time. Stabile's claim was corroborated in 2009 with the discovery by Jorge Romance of a similar use of the symbol in Castilian documents from the 1440s, referencing an arroba of wheat. As noted in this latter link, the symbol has also been seen in one earlier occurrence in a 1345 Bulgarian translation of a Greek chronicle, where it merely substitutes for the "A" in the word "Amen." This use has apparently not so far been connected to the others (and may have been merely a scribal ornament around a letter of a prominent word).

Thus, Stabile's theory postulates that the @ symbol originally came into use as a Spanish and Italian abbreviation for the standard measurement units of arroba or anfora, which explains its name in modern European languages. The use apparently spread through Southern Europe, where @ eventually came to stand in for a generic mercantile unit, leading to our modern use of the sign to mean a sort of "units at [price]."

Stabile's theory doesn't explain precisely why a circle was initially used around the "a" in the abbreviation, though Romance's manuscript findings seem to show an earlier non-standardized use in the 1400s where the "a" was simply ornamented in various ways (including incomplete circles). But the corroboration between modern language terms for the symbol and historical manuscript evidence makes a stronger case for this origin than any competing theories so far.

Just as an addendum, I assumed the question is mostly about the early history of the symbol before it was appropriated into e-mail addresses and such. But since that meaning is also discussed in the question, the story of how @ was incorporated into typewriters and then was chosen by Ray Tomlinson as the standard e-mail symbol is discussed in the first link above and its counterpart here.