Cwadda's not being glib. Nobody can answer this for you - emergency funds are loaded with the appropriate amount of money that leaves you stress-free about needing money set aside. For some, this has to protect rental assets or homeownership or whatever, too.
I have like 25k in credit with balances paid off every month that never exceed 1-2k. I don't keep more than 100 bucks in cash and have no stress whatsoever.
Some people need 2k.
Some people keep 30k.
If you can't sleep without 30k in checking, that's an awfully steep price, but you're going to be best able to determine what your threshold is.
If you need someone to say "it's not wildly irresponsible to use credit for your emergency fund," it's not, IF...
1) Your monthly cashflow would solve most emergencies by just not funneling into savings
2) Your credit would cover any gaps in that available cashflow or any immediate needs without racking up balances (could be paid with monthly cashflow before interest dings)
As for timing, 15 days of interest on one-month's savings is not a significant cost if you're getting peace of mind out of it. I get paid similarly and deduct on the 5th, because I want that savings to go up ASAP. If anything, I get stress from waiting for it to go up, not from lacking an emergency fund.