The option to change the MC's name comes about in the common route as you're playing. When the MC decides to use their middle name as a pseudonym, you get a choice between "Cora" and "Something else." Choose "Something else." to do a custom input.
If your version of the game was the original one released, it defaults to NSFW and there's no option to turn on SFW mode. If you have the version that was just released (V1.2), there will be a choice screen after you select New Game that asks if you'd like to turn NSFW on or off. So if you don't see that screen, then you have the original version with only NSFW content. :)