Q:

When did "phone" become accepted as its own word? 

In older print publications, I have come across telephone shortened to 'phone, with an apostrophe to mark where the beginning of the word had been omitted.  Now, however, phone does not need an apostrophe and is viewed as its own word, spawning other compound words like phone call, phone line, and phone book.  
When did phone start to replace 'phone?  
Is there a term for the phenomenon of an abbreviation becoming a word in its own right?  I know something similar happened with facsimile and fax. 

A:

When did phone start to replace 'phone? 

Immediately, if not before. 
Let's just briefly answer the second question and then come back:

Is there a term for the phenomenon of an abbreviation becoming a word in its own right?  I know something similar happened with facsimile and fax. 

Clipping.  Words get shortened and lose rough edges like pebbles in a stream. 
But, let's consider that we're in 1880.  While there have been several devices called "telephones" since 1828, it is only 4 years since Alexander Graham Bell's device of that name for transmitting voice communications have become a practical reality.  They are a novelty, and their name is long, and these two things encourage shortening. 
Now, we hear someone say /fəʊn/ or /foʊn/. Do we consider that a contraction, an abbreviation, slang word, or a combination of two or all of those? 
There really isn't a rule we can follow.  If we consider it a contraction, then we would write it as 'phone.  If we consider it an abbreviation we would write it as phone and if we consider it a slang word we'd likely (this being the late 19th century when people put slang words in quotes) write it as 'phone'. 
All three are found:

1880   Decatur (Illinois) Daily Republican (Electronic text) 27 Jan.   Haworth & Sons were among the first to subscribe for a phone, but the company was a little tardy in putting in the instrument. 
1886   California Maverick (San Francisco) 13 Feb. 1/3   To him I related the famous fiend's new invention—this 'phone that could talk in foreign languages. 
1899   Westm.  Gaz.  18 Apr. 2/1   The receiver of this ‘phone’ consisted of a horizontal cylinder divided vertically by a diaphragm which projected several inches beyond the front orifice. 

Our choices would vary according to how often we heard talk of phones, and our individual attitudes toward contracted speech, slang, and innovations both linguistic and technological.  But from the very beginning there were some accepting phone as an independent, if perhaps rather hip word. ( A very hip word, since hip wasn't to acquire that sense until the early 20th century). 

A:

The particular form of word formation is known as clipping. 
The Wikipedia article is informative, but I'd say needs refining.  It does contain a good sub-classification and quite a few examples.  And also the definition:

In linguistics, clipping is the word formation process which consists
  in the reduction of a word to one of its parts (Marchand: 1969)

Choster's comment seems the best response to the other question, although it would seem to suggest that 'phone appeared later than phone and then largely disappeared.  Like him, I would not post the link to Etymonline in an 'answer' as I wholeheartedly agree with what Martha says on Meta:

‘If we accept the axiom that trivial questions are bad for the site,
  then the proper response to a general reference question is: 1.Don't
  answer!  2… [Vote] to close.  The point is not to encourage trivial
  questions.  If you post an answer or vote up an existing answer, you're
  implicitly encouraging more questions like it.’ – Martha

The snag here is that OP's other question is neither general reference nor trivial. 

A:

It happened with omnibus > bus; taximeter cabriolet> taxi; refrigerator > fridge; and other such colloquialisms.  The apostrophe was used to indicate the omission of a letter or letters, then eventually dropped when the word came into more colloquial use, probably because it is not verbalized.  
According to Google ngrams the use of 'phone' overtook* the use of 'telephone' in early 1990, at least in print. 
* thanks to edwin ashworth

