I think the first thing you need to do is have a chat with nursery. They generally are very experienced in dealing with tantrums etc. but as the behaviour you describe potentially goes beyond the normal foot paddling tantrum of a 2 year old I think you need to meet with them to get a full picture of what is going on. You need to know frequency and scale of bad behaviours, triggers, times of day, types of behaviours etc. You need to hear whether her issues are within the normal range or not and, if not, what support (if any) the nursery can provide. If her issues go well beyond normal you may also want to speak to your GP about it too.
The fact they have spoken to you about their concerns suggests that normal behaviour management strategies are not working. That doesn't mean, as some people are suggesting, that the nursery is necessarily poor (and you should have a general idea about that anyway). It may mean that your DDs behaviour is beyond what is normally expected and observed in nursery at this age. Whilst all the behaviours you describe are 'normal' to an extent, what may be abnormal is the scale and frequency of your child's behaviour: this is what you need to try to establish. Does she have social difficulties? Would she be better having the routine of full time in nursery rather than part time. Having found that a monumental tantrum gets her home earlier to mum, is she now more likely to repeat this behaviour?
You need to discuss with them the patterns and triggers for behaviour. If she is forgoing her nap time (as mine did at that age) then this could help explain her afternoon behaviour: she is over tired and over stimulated. Was quiet time with a book appropriate - for her age, yes; but with her behaviour problems did they set her up to fail on that one? Are there any particular foods at lunch time - or E number juices on the table - which seem to exacerbate her behaviours? What behaviour strategies are they using: what has already failed and what are they trying instead?
You need to try to do the same at home: what triggers the worst behaviours: and prioritise what you want to tackle now. If bed time is the main issue now, change your bed time routines. Have DD go at a different time from the 4 year old. Have a story time and cuddle together. Make sure she has all her needs before she goes to bed: toilet, supper etc and a bottle by the bed to minimise excuses for getting up. At this age my eldest still loved a bottle of milk at bedtime: and that certainly helped her drop off too. Don't use the 'dark corner' as a threat - you will make bed time associated with fear and punishment when you want it to be something nice - her time with your attention. Have a reward chart for staying in bed to earn iPad time. Don't let her play stimulating games - or really any ipad activity - before bed - it is not restful and hypes them up. As for the screaming etc. I have found telling my barrack room lawyer she can scream all she wants over there rather than next to me soon decreases the behaviour! Consistently sticking her in timeout away from you too - and consistently putting her back if she moves - will eventually see results. Give her one warning, then put her there.
Potty training - if you are using them, do away with pull ups as they slow the process down by removing the unpleasant sensation of being wet. And also remember it is normal for a child this age to still get so absorbed in what they are doing that they wet themselves. Like you say, it is not a biggy at the moment.
I would say your most immediate action is to arrange a meeting with the school so you can develop a plan to support your child that works for you and for them. Whilst many 2-3 year olds will tantrum at school obviously you want her to learn to be calm and not react physically to others when she can't get her own way. I do wonder if she is getting frustrated because she feels she is not getting listened to/getting enough attention at school. Being disappointed over not winning needs to be acknowledged - it's ok to be mad and sad your team lost. If she feels they were simply ignoring this feeling by telling her to be a good loser she may simply have lashed out in frustration. Likewise destroying the book - got her some attention I expect and got you to come to school. If nap time is when teachers take their lunch then she may have felt very sidelined. These are not excuses for her behaviour - but you do need to think about the purpose of her behaviour, because she certainly has one. If you can figure this out, then you and the school can work together to crack the problem.
Sorry for writing an essay :)