There are basically two steps in this: 1) You MUST prove you have the required money/income, 2) You MUST comply down to the tiniest detail with the list of required documents...
, does not seem realistic. Whenever you apply for an extension, they will add it to your current admitted-to-stay date (or a little later) and ask you to come and get it while you are not in Thailand. So when you return to Thailand you do not have a valid visa any longer. Then you will have to apply for a new extension. In theory, you can get an extension based on any legal stay in Thailand if you meet all other requirements. However, I would ask my local immigration about how to do this most easily. If they give you some "attractive" advice, then make sure to get it in writing as such "favour" most likely will be officer-of-the-day-granted...
, of course, your name needs to be listed as her father. The legalizations are standard procedures for any country. If you are lucky to get the passport, then you/she might very well run into problems later on. If the translation is not 100% correct and not legalized, then you are submitting false information.
You will need a certified translation of the Thai birth certificate into English. You can get this at the Ministry of Foreign Affairs. Then you will need to get it legalized at the Ministry of Foreign Affairs. Finally, you will need to get it legalized at the British Embassy. Now you bring your daughter along with the legalized translation to the British Embassy and apply for British Passport for her.
I have been advised by my immigration that the online systems are still not working properly. If it works, you can use it. If not, then you must show up at the immigration.
, it depends on your visa. For some visa types, yes, like visa extensions. For others, the admitted-to-stay date will be updated. This counts for Elite Visa and others. For visa extensions, your next-notification-of-address date will be updated.