You don’t have to enter any earlier than the expiry of the visa. You could enter on the last day if you wanted, but that would get you a single 90-day stay out of a 1 year ME visa.
Is her situation risky enough to warrant the 5000 baht fee, short overnight trip to Savannakhet, queue to submit application and collect visa the next day?
You should have no problem entering on visa exemption every 4 months for stays not exceeding 4 weeks. Save applying for a non-immigrant visa for later when you want to live here more permanently.
No, on your entry this Thursday, you will be given 90 days to stay (stamped date will be in September). When you leave on 13 June, getting a re-entry permit will allow you to come back and resume that permit to stay. You will not start a new 90 day stay. But that means you have until early September to get your next visa/extension.
When you enter this Thursday, you should be stamped in until September. Get a re-entry permit when you leave on 13 June and you can still re-enter Thailand and keep this permit to stay. If you are traveling often, get the multiple type. That will give you a lot more time to arrange for your next visa, or if you meet the requirements, an in-country extension of stay for 1 year.