The part number of the carb you bought on eBay is indeed for Honda GCV160 engine, but it is not the right one for your mower. Not saying that is what the issue may be, but you really should get the correct carb installed first. The correct Part Number is 16100-Z0L-023, list price is $15.08.
Google the part number to find a Honda Dealer selling it online, or use this link to find a Honda Dealer in your area:
Find A Honda Dealer
With the correct carb installed, do a spark test first, and confirm you are getting good spark.
Drain ALL the fuel from the tank and fill with fresh 86 octane regular unleaded.
Next, make sure the choke is working properly; it should fully close the choke butterfly when engaged, and fully open when the throttle is moved to FAST.
Check the air cleaner element for rips, tears or excessive dirt, clippings, debris.
Confirm all the gaskets and spacers are correctly installed with the carb, and all spring, linkage etc. are also correct.
Make sure the two long bolts that hold the carb, choke and air cleaner are secure (vacuum leak can cause surging).