Get partial match from GIN indexed TSVECTOR column
Do all queries have to be be in dictionary?

No. Because only word stems (according to the used text search configuration) are in the index to begin with. But more importantly:
No. Because, on top of that Full Text Search is also capable of prefix matching:
This would work:
SELECT id, subject
FROM   mailboxes
WHERE  tsv @@ to_tsquery('simple', 'avail:*')
ORDER  BY id DESC;
Note 3 things:

Use to_tsquery(), not plainto_tsquery(), in this case because (quoting the manual):

... plainto_tsquery will not recognize tsquery operators, weight labels, or prefix-match labels in its input


Use the 'simple' text search configuration to generate the tsquery since you obviously want to take the word 'avail' as is and not apply stemming.

Append :* to make it a prefix search, i.e find all lexemes starting with 'avail'.


Important: This is a prefix search on lexemes (word stems) in the document. A regular expression match without wildcards (content ~* 'avail') is not exactly the same! The latter is not left-anchored (to the start of lexemes) and would also find 'FOOavail' etc.
It's unclear whether you want the behavior outlined in your query or the equivalent of the added regular expression. Trigram indexes (pg_trgm) like @Evan already suggested are the right tool for that. There are many related questions on dba.SE, try a search.
Overview:

Pattern matching with LIKE, SIMILAR TO or regular expressions in PostgreSQL

Demo
SELECT *
FROM (
   VALUES
     ('Zend has no framework')
   , ('Zend Framework')
   ) sub(t), to_tsvector(t) AS tsv
WHERE tsv @@ to_tsquery('zend &lt;-&gt; fram:*');


 id |       t        |          tsv
----+----------------+------------------------
  2 | Zend Framework | 'framework':2 'zend':1


Related answer (see chapter &quot;Different approach to optimize search&quot;):

How can I generate all trailing substrings following a delimeter?

Emails?
Since you mentioned emails, be aware that the text search parser identifies emails and does not split them into separate words / lexemes. Consider:
SELECT ts_debug('english', 'xangr@some.domain.com')


(email,&quot;Email address&quot;,xangr@some.domain.com,{simple},simple,{xangr@some.domain.com})


I would replace the separators @ and . in your emails with space (' ') to index contained words.
Also, since you are dealing with names in emails, not with English (or some other language) words, I would use the 'simple' text search configuration to disable stemming and other language features:
Build the ts_vector column with:
SELECT to_tsvector('simple', translate('joe.xangr@some.domain.com', '@.', '  ')) AS tsv;