Can you initiate a challenge type (phase 1 of framework action) and decide to kneel no one in this challenge (phase 2 of framework action).

If yes can you stil continue till determine winner of challenge or does it stops immediatelly without abilit to play an action (like using the new lanniter location that put a attacking clan in the current challenge or the big stark that jumps into challenge).

You must declare at least 1 attacker when initiating a challenge.

The rules state that you must do both. You cannot initiate a challenge without declaring at least one attacker. The two steps in the framework action window are not mutually exclusive. If they were, you could say "I attack with Tyrion" without naming a challenge type.

Is the declared attacker required to have strength greater than zero when declared? i.e. can you declare a strength zero character as the sole attacker.

No, you're only required to declare an attacker. Its strenght is not relevant when you declare it.

Of course, you cannot win the challenge without counting at least 1-STR during challenge resolution, but STR doesn't matter at all when you are simply initiating the challenge.