How much below asking price can I offer?
It depends on the comps and the market. If similar homes sold for less, or the home has been listed a long time, an offer below asking can be reasonable. If it is priced on target and competitive, a low offer may be ignored. ListingCraft for Buyers's price verdict (Priced high, On target, Priced low) tells you which situation you are in.
Explain your price with the comps, and keep other terms clean, such as a solid preapproval and reasonable deadlines.