I have had two extractions so I have some experience here.
As with most things at the dentist, the anaesthetic injection is more painful than the procedure itself. You'll be numbed up in such a way that it just feels like someone pressing very hard against your jaw. One of mine was so easy I had no idea they'd finished, the other, (a deep filled molar like yours), took a bit more effort but was not otherwise painful. For that one they were worried a root might remain, but even with the poor state of the tooth it all came out in one go, phew. I imagine everyone is different, but I did not find that the teeth either side shifted very much to take up the gap.
Both healed up fine. Just make sure you follow all aftercare advice to the letter.
In terms of overcoming nerves, I would just keep reminding yourself that some temporary discomfort is much preferable to being in excrutiating pain long term.