Correct, if you enter before 6 Jun than you get 60 days of stay, if you leave end of Jun you can apply for the re-entry permit and when you return you get the same Admitted until date.
From your girlfriend as her name is on the contract, but you might ask the landlord. As long as you have a copy and an invitation letter to stay there.
No matter what, your address should be registered at the local Immigration Office within 24 hours after arriving, if done in person the form needed has number TM30.
For your Extension application I would advice the lease or rental contract and a letter from your girlfriend that she invites you, with full name and passport number to stay with her full name and id card number at this address.
If you convert your entry into a Non Immigrant Visa at Immigration you get a new 90 day stay without entries.
If you wish to travel during these 90 days you can apply for a re-entry permit and on return you will get the same last day as Admitted to sta. You can than apply for the 1 year Extension of Stay.
Or you leave the country, apply for a new Non Immigrant Visa come back, get 90 days on entry and apply after 60 days for the 1 year Extension of Stay.
Once the 1 year Extension of Stay is stamp in your passport you need to report if you are 90 days inside Thailand.