Please join the attached the group. They can help you with requirements to get out philippines... Which is much hard than getting into Thailand. You will like need to complete CFO, and will need authenticated AOS from your boyfriend, plus a stack of evidence of who you both are and your relationship.
You probably misunderstood. The 60 days is for valid of entry, meaning you must enter from 60 days from the date of issue. When you enter you will get 90 days. If you do not enter within 60 days (dates are listed on the visa) the visa is void.
As far as how long it takes it will very based on the embassy and how busy they are. I did a STV last year in Taipei and it took 1 day, but that's probably not that common.
I don't have a school like that to recommend, but that is visa mill type school and not really going to learn Thai. 4 hours per week does not meet the requirements for a visa. The minimum requirement is 100 hours per 90 days. Be aware of schools offering that is not actually legal.
You can come now. Thailand had been open for tourists since Sept 2020 and never closed since. With your Australian passport you can come visa exempt or buy a tourist visa. You can enter via phuket sandbox where you have to stay on the island of phuket for 7 days if you are vaccinated. Otherwise hotel quarantine in bangkok.