'Well we usually attempt to calm her down from being in the midst of a meltdown by cuddling her and explaining to her that she needs to behave better and why what she doing is not good or allowed.'
I think that's your problem, she gets far too much attention for negative bheaviour. You need to ignore it. Then when she has calmed down and is behaving well, cuddle her and tell her how much you like what she's doing now. Instead of constantly telling her why what she's doing isn't good, focus on telling her why what she's doing is good. Stop criticising her, she's 4.
Things like 'chattering incessantly' isn't naughty, it does drive you mad, but it's just what kids do. Equally pulling faces and messing around isn't naughty. There's not really anything on your list that is 'bad behaviour' it's just 5-year old behaviour. Instead of a an hour long lecture, you need to ignore and distract.
What happens with going to the toilet if you don't nag her? Does she actually wet herself, or will she go just in time?
If she doesn't want to watch TV, then fine! Don't mahe arguments where there doesn't need to be any.
Being stroppy and saying she's going to kick you or whatever, all normal when in a temper tanturm as well. Leave it at the time and then when she's calmed down, tell her briefly 'that isn't a nice thing to say' and encourage her to say sorry (don't force the issue though).
'He spent an hour explaining to her that she needs to be good and not make us sad'
This just makes me
. She's 4. It was bedtime. She's probably tired.
How can she be so good at school and so annoying at home?
From the sound of it, because they handle her beahviour significantly better. The opposite to good isn't annoying btw. She's probably annoying at school too. 5 year olds can be 'annoying'. Maybe the teacher has more realistic expectations of 5 year olds and a higher tolerance to deal with their constant chatter/messing about/stroppiness etc.
Have you considered a parenting course? I think they can be really helpful for giving you strategies to deal with this sort of thing.