For H1B costs, please read the redBus2US blog H1B Visa 2013 filing Fee ? Should you pay for H1B cost?
If your petition is filed under regular processing, it could take 2-6 months for a decision. Whereas, if your petition is filed under Premium Processing, it will be processed in 15 calendar days.
Yes, a different company can file cap-exempt H1b petition on your behalf.