Am I at fault? Can I get my security deposit back from Landlord?
The landlord is wrong, and he should reimburse you the totality of the security deposit.
Clauses #1 and #16 would entitle the landlord to deduct from your security deposit only the portion of the 7-day period that is not already covered in the payment you made for 1st week's rent, which according to clause #1 starts on Sunday July 29.
If I understand correctly, you signed the contract (and paid) on July 27, and then gave your 7-day notice on July 28. Since you paid an additional amount for Saturday July 28, then your 1st week's rent actually covers the entire 7-day period of your notice. Therefore, the landlord should reimburse you the entire security deposit.
The contract nowhere indicates that the count of [post-notice] 7 days starts after the current rent-week elapses. But even if it did contain language in that sense, the dates you mention render that hypothetical clause void.