How can I ensure my ED non-immigration visa is processed in time for my internship in Thailand, and when should I go to the immigration office after arriving?

I start with the last question first...

The validity of the visa is the time period when you have to enter a country. In your case that means 90 days from the date when the visa is issued.

When you arrive in Thailand you will get an entry stamp (not the same as a visa) giving you permission to stay for 90 days (the day of arrival is day 1).

You can most likely go to the Immigration Office to do an extension when you have 30 days left of your initial 90 days (some immigrations allow earlier and some later, if you want to know for sure, check with the local immigration where you will be staying).

As for, if you will get your visa in time, that depends on WHEN you have all the documents needed to submit your application and WHERE you will be when you apply. Even though all applications are done online, the estimated processing time varies depending on which embassy/consulate you're applying through. I suggest that you read the information provided on the web page for the Thai Embassy in the country that you will apply.

As long as you get the required documents and make your application soon, I'd say you'll get the visa in time (most embassies process visas in less than a month).

You won't get the visa without all the required documents