M.,
Hi, i can relate somewhat. My daughter is a big whiner, but doesn't have the eczema, my friend's son has it bad though...she says that the dr. had given her some kind of cream for him, and it didn't really work all that well. she said that instead, she pretty much covers him in a petroleum jelly like product. the sun is GREAT for eczema, unfortunately, heat is REALLY BAD! as soon as it starts getting cooler out, you should do what you can to have him outside as much as possible. until then you need to try and get him out in the sun during any possible cool points of the day (even if it has to be early in the morning as soon as he gets up). try not to bathe him too often, especially not in really warm water. and if you can, put anything cold/cool on him to help ease the pain...if she's shopping and he starts itching, she takes something from the frozen foods and puts it on him to help soothe the pain.
as for the whines and screams...keep doing what you're doing. console him when he's not feeling good, but if he's fine, put him down when he starts whining and keep telling him that this behavior just isn't accepted. when my daughter starts i'll pick her up and tell her how much i love her, but that she needs to learn and play by herself. yeah, she likes to just sit on my lap and watch her morning (educational) cartoons...but if i just sit on the floor with her and get her preoccupied with playing, i can sneak off and do some cleaning.
don't listen to people who tell you that you need to be a mother when you're with him...YES, you HAVE to be his mother, and you HAVE to spend your time with him!!! BUT...you do need to get things done too. try to get as much done as possible when he's sleeping, but what you can't get done then, do while he's playing. it takes some time, but by when my daughter was not much older then he is now...she started going off and playing on her own. but for some reason she just wouldn't do play on her own (or let me leave the room w/out her freaking out...even if she could still see me in the other room) until she was able to walk without help. the reason he probably acts so well when at daycare and not at home is because he was colic and is used to you pretty much dropping everything for him (because of the colic and wanting to soothe him) and he's used to it, but no one at daycare did that, so he doesn't expect it from them. kids have the amazing ability to pick out the saps that they can twist around their lil finger and play them until they get what they want. unfortunately mommy is usually the biggest sap of all (i KNOW i am!!!) even if you can't stand it, and it drives you crazy, you don't want to hear them whining and screaming...you want them to be happy and can't help but give in to see them happy. once i learned that i just tried to always turn her attention to something else. she's MUCH better now...the only time i really have the problem is when she's really tired, sick, or teething (thank god that she's almost done with that!)
if you need to clean, and can't seem to get him to play on his own. you can try what my mother does (which SOMETIMES works for me). you can try to give him something to clean while you clean. my mother likes to just tell her to help and put things away, while a friend of mine likes to tell her nephew to help her clean, and hands him a sponge, and he'll just go around and wipe everything down...even the walls lol! when i get her to help me, she'll put some things away, but she likes to hand me things. i may have to repeat myself a ton of times, but she gets all excited and runs for what i ask her to get, and runs back to me with it with a huge smile...then i clap my hands and say "oh, you're such a big helper for mommy...THANK YOU!" doing this not only gets them to start learning to clean (and making cleaning as fun as possible works the best), it also helps get done what YOU need to do while spending time with your child!
if you can't do that just yet (if he's not really walking yet) don't worry, just be patient and try to get him to play on his own for a little bit at a time until you can try getting him to follow you and help out (once starts to walk, if he's not already). also, i know people say that children so young shouldn't watch tv, but if you find something educational to watch, you can put him down to watch something (like sesame street, little einsteins, mickey mouse club house, etc.) and this way he's not only preoccupied with that, he's also learning! i do that in the mornings. it's MY background noise (since i don't like things completely quiet) and her learning, and distraction while she eats, plays and i clean or do what i have to do. once i'm done with as much as i can get done at that time, i try to sit down with her and continue teaching whatever she learned while watching tv, then it's nap time and i continue doing what i have to do. well, i HOPE i helped even just a bit...GOOD LUCK!