moving around is good, but every time you reside at an address your address must be registered within 24 hours. Every hotel ask your passport to make this report.
You can apply during the last 30 days BUT the 30 days under consideration is part of the 1 year Extension of Stay and added to your currect Extension.
You can travel during these 30 days BUT need a new Re-Entry Permit after the under consideration stamp is placed in the passport to get the same end date.
And you need to inform your local Immigration Office about your travel plans, so they can organize the possible house visit.
Look at their website, have all requirements in place and it should be easy. That others prepared correctly and got the visa can not be used as argument if you did not came in as preperad as them.
I removed a discussion about the difference between the e-mail or sticker you buy BEFORE you make your way into Thailand and a BIG STAMP you get from the Immigration Office AFTER you entered Thailand.
I fully understand we call it all the same to make live easy but to give advice, we need to use the correct terminoligy.
Like you are in the bar, you do not accept a lime juice if you order your favorite whiskey and get told you can not read, this is what you ordered.