You are asking four Questions in One, "SMART" of You.
Your each Q u e s t i o n is being replied in same sequence;
1. Avoid two week experience, in your resume.
2. Reason/s for a gap, which is acceptable to interviewer can be any or in combination for which you can convince conveniently.
3. Not all, who are not verifying the credentials, of perspective employees.
4. You have experienced and realised, that's why you should certainly avoid stating two week experience, in your resume.
Choice is yours.
Good Luck ! !