Can I give you a small piece of advice here @Abitboring - just don't react to the more vicious posters? Some people here seem to get a kick out of being as nasty as possible, and it's not worth rising to their bait.
Having said that, it's kind of coming across that you dislike this new hire, and I hope you are being more patient with her F2F because you seem somewhat prickly. I understand that you are under a lot of pressure.
No employee can always be on annual leave because it's limited. Your problem here is your manager, who should be ensuring that you get fair and equal access to annual leave, and that each of you is available to cover the other. In my team, for peak holiday periods, everyone has to submit a request to the team manager, not their direct line manager, who has oversight and the information available to ensure that cover is always provided. In practice this has meant that things usually do work out and leave is seldom denied.
I think the ideas you've had about creating a training manual etc and encouraging her to take ownership of her work are all good, and should help. If she doesn't cut the mustard, then you may well have need of the manual again in the near future. I'd also suggest that you set up a training schedule, formalising arrangements for say 3 months at a time, and then you can see from the outset whether or not she is going to be available - reducing your frustration and meaning you can plan ahead more effectively.
Kindly, I also think you need to take some time out for you, because it sounds to me that you are under stress and that your mental wellbeing is suffering. Does your employer offer any EAP/counselling support? If so I'd encourage you to use it and either way, I think you would benefit from a chat with your GP.
It's that old chestnut of putting on your own oxygen mask before trying to help other people. You need to look after you - because, trust me, no-one else will. Best wishes.