Yes, you could insert a comma after 'skills' or, since it's a headline, you could simply say:

"JAVA developer with excellent analytical skills available for new challenges."

Personally, I would say "seeking new challenges" rather than "available for new challenges."

Yes, you can and should for two reasons:

1. Contraction

When we form a contraction by removing letters in a word or between words, we use an apostrophe. When we form a contraction by removing entire words in a sentence, we use a comma.

2. Absolute Phrase

When you trail a sentence, clause, or phrase with a modifier that modifies all of the preceding, the subject (or in this case, what would be the subject if you were to say "is"), or the noun at the beginning of the phrase rather than the most recent noun, we use a comma.

In what you've written, not putting a comma there make "available for new challenges" modify the noun that appears immediately before it, "skills." To have "available for new challenges" instead modify "JAVA developer" back at the beginning, you need a comma there.

In Journalese writing, yes. In academic writing, no:

"JAVA developer with excellent analytical skills, available for new challenges."

In academic or formal writing, the above are sentence fragments with no overt verb in either "clauses":

"JAVA developer with excellent analytical skills is available for new challenges."

Note: the part of speech of "available" is not a verb in the above, but an adjective. It follows the structure of adjective + preposition:

I’m surprised at how fast my students are learning.
My mother is angry at me because I forgot her birthday.
Jamila is good at songwriting and painting.
He’s terrible at math – he failed the class twice!

ODO also cites "available" as only an adjective, especially in the example:

‘the nurse is only available at certain times’

In Journalese writing and in informal contexts, sentence fragments are usually accepted.

This, of course, depends on the style guide you consult. I'm not sure whether it is a convention in Journalese writing to omit the main verb; however it certainly is not impossible. I've seen a few headlines without a verb:

Earthquake and Fire: San Francisco in ruins
North Korea trip 'successful'

This specific style guide states:

Subject and verb, please: Don't write headlines in which nouns and verbs (other than “is” or “was”) are assumed. And don’t start a headline with a verb. (See Problem Headlines section.)

Use the active voice: Effective headlines usually involve logical sentence structure, active voice and strong present-tense verbs. They do not include “headlinese.” As with any good writing, good headlines are driven by good verbs.

Problem headlines

Example #2: Police chase winds through three towns

Huh? Are “chase” and “winds” verbs or nouns?

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Other ways you could write your headline with style:

1. "JAVA developer, excellent analytical skills, available for new challenges."
   
2. "JAVA developer with excellent analytical skills; available for new challenges.

or if "excellent analytical skills" is a non-essential appositive element in your clause as in (1) it can be omitted:

Java developer, available for new challenges

or

Java developer available for new challenges.