Really sorry to hear things are not happy there.
I really wouldn't expect a 1yo to wake from hunger, there are a dozen or more other reasons their sleep may be disturbed but hunger is not one I would bet on (not unless a LO had been very ill and was still in a phase of recuperation and needing to make up for lost calories from their period of illness).
We always have sleep disturbance around the birthday which lasts several weeks, do you think it could be this? His mood in the day too could be down to developmental leaps at this birthday mile stone. We get the behaviour/mood every birthday and half birthday.
Is it okay to give an egg every day? Do I not need to be concerned about too much cholesterol?
Unless there is an egg allergy/intolerance in the family or for LO then it's fine. Eggs are 'good fat' or 'good cholesterol' and are no longer considered to be something which needs to be limited to one or two per week.
LOs also need more fats than most adults which is why they have full fat milk and cheese etc and lower/reduced fat foods are not recommended for 1yos.
I've never had a problem cleaning the sippy cups myself, I get them into hot hot water and use a small bottle brush but I know what you mean, my mum once 'washed' some sippy cups and I saw the sippy holes clogged with stale milk where they had just had a quick dunk rather than a scrub. It's ok to put them in the steriliser you know, along with the bottles.
Have you tried some different foods at breakfast (even if not eggs)? Have you tried bringing meal times earlier?
Do you feel you have a good grasp of portion size for toddlers, I'm just wondering if he's just not hungry and you trying to get him to eat is frustrating him?? There was a marked drop in appetite when mine turned 1yo, I probably had a thread about it too! It's quite common.
I'll keep having a think xx