if you apply from a Thai embassy you can get the 90-day visa which will let you open a bank account once you arrive in Thailand. And that will let you prepare to apply for the 1-year extension.
it's easier at home than it is in Thailand. Especially since you have to have a Thai bank account or embassy certified income to apply in Thailand and you cannot get a bank account without already having a long term visa. So that means agent is the only option for the bank account
There is no limit. But that doesn't mean it's unlimited. It's much more about how long you've been in Thailand than it is about how many times you've entered. Around 90 days in the past 6 months they start to interrogate you. Around 5-6 months in the last year or so and you're looking at possibly being denied entry. A tourist visa isn't likely to help if it's gotten to that point.