The Department of Labor does not charge filing fees for either of these filings.
The ETA-9035 Labor Condition Application ("LCA") normally takes about a week from the time it is filed. However, there are often issues with the employer's tax ID number not showing up in the DOL database causing denial of the LCA. If no LCA has been filed for the employer since the iCert portal went online, it is advisable to pre-clear the employer's tax ID using the DOL's procedure for doing this before filing the LCA - the procedure normally takes one to four days.
The ETA-9089 processing time can vary widely. Right now, some cases are coming back approved in as little as three months. If the case is audited, however, processing time can be well over a year.