Can I extend my stay in Thailand after arriving on a 45-day visa exemption?

if you're from one of the countries that gets a 45 day visa exempt entry you can show up, get stamped in for 45 days and then apply for a 30 day extension for 1900baht once at the immigration office.

Now whether you need proof of onward travel to check IN to your flight here flying without a valid visa in your passport (coming in visa exempt) is between you and your air carrier. That's who you need to check with

You won't have any issue once you land here at passport control with onward travel 55 days out when you stamp in.

yep, 45 on entry and a 30 day extension