Generally your purchase contract will state in a firm dollar amount how much the seller is paying towards your closing costs and prepaid items. Example - $4000 or 3% of the purchase price
The agreement rarely would be so vague as to say they'll pay all your closing costs - it will be more specific than that.
With regards to the appraisal itself, most lenders will not let that be paid for out of escrow because the appraiser requires payment up front and if the loan doesn't close then the seller is not going to pay for that for you. No one will get $375 back, but at the time of closing they will probably take $375 that could have gone towards the appraisal and use it to pay something else like some of your tax escrow or something.
Hope this clarifies things a bit. If not, press your lender for more clarification. Best of luck... more
Short answer yes. The bank statements will show the deposits each month into your account, along with outflows. Sometimes the underwriter will catch child support, other payments that may be missed on the mortgage loan application, that will result in a turn down of your loan.
Also, if you have deposits going into your account that cannot be sourced, this will result in the assets on your application to be reduced by the amount not sourced and seasoned by a eligible source.
Lastly, if you have bounced checks or nsf's fees, this can result in a turn-down of your loan because this is a sign that you cannot handle your money.