Do I need to show proof of residency while living with my Thai girlfriend for a retirement visa in Thailand?

Your 1-3 are all wrong as you cannot get the 90 day Non-O (retirement) Visa in country without a Thai bank account. You can get your Non-O from your "home" before travelling to Thailand without a Thai bank account.

She needs to complete a TM30 for you

It really depends on the branch and the day. Go try the same bank she has with her and her house book and see if you can do it without.

My sister-in-law is my TM-30. You can do it too. She comes with me to immigration to fill out the form.

You can live with her. I just done for 5 months.just show copy of her blue book. I just took her to immigration office with me. And blue book. All was fine.

You may need a letter from your girlfriend indicating you are living there. You should go to your immigration office well before you plan to apply to ask for their list of requirements for the non-O visa and you can ask them what proof they would need for you living with your girlfriend. It may involve an invitation letter and a signed copy of her ID card and/or housebook.

To convert to the 90 day non-O visa you must have the 800,000 baht in a Thai bank account in your name only. The money must be shown that it was transferred from outside of Thailand. Most immigration offices do not require this money to be in the account for any amount of time, just that it's in the account with the required proof, but some immigration offices do require that it has been in the account for 2 full months when you apply (It's not 60 days, it's 2 full months)

After getting the 90 day non-O visa you can then apply for the 1-year extension after 45-60 days depending on the immigration office. This is the part that requires the money to have been in the account for 2 full months at all immigration offices.