I don't have a baby, but your concern for clean hands is very reasonable! I also have to tell SD12 to wash her hands after using the loo! Why can't 12 year olds do that on their own??! How often is he with you? Your husband really needs to be backing you up when SS ignores you or doesn't do what it asked of him.

Ew, NO... Babies are VERY susceptible! If he doesn't wash his hands on FRONT of you, you don't have to let him touch your baby...

lay this out as you intend. It's disgusting not to wash one's hands. It's something this kid should have already covered in health class through school.... but kids are kids and they need to be politely reminded. When that doesn't work, it needs to go to a strike three, you're out, level. Your SS obviously is at the least, strike two.

So now he needs the sterner warning. Listen up kid, keeping your hands clean, especially around a baby and when using the kitchen area are not optional. It's mandatory. 

Set up a handwash station, not just in the bathroom but put liquid bottled soap and a hand towel in the kitchen. Then lay out the expectation along with the 'why' (don't do the old 'because I said so'). If he (or the younger brother) want to play with and/or come around the baby, they go wash their hands. Let their father set the example by having him be the role model ( aka, Dad immediately starts doing what is expected, he washes his hands prior to playing with baby ect). Doesn't matter if kid washed his hands after the bathroom already, hands spread germs and one can pick up germs on one's hands anywhere at anytime. Have him sit down and look up importance of brainwashing online. This isn't you being petty or being a nag just to nag, nope. It's real and it's important. AND you expect your DH, their father, to set the example.  

Then lay out the 'clean up your messes'. Yep, the whys too. Not only are you not their maid, but baby is going to begin start being more mobile. Baby puts things in their mouth. Baby will be on the floor crawling or just playing on a mat. Next baby will start to learn to walk and that takes a clean area. And don't forget the speech about how everyone living in the household has responsibilities and expectations. It's a normal part of being and living as a family. 

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I would be bothered... I love hygeine and nutrition though... So a lack anywhere really bothers me! Like when I walked into the bathroom and SD9 was brushing her teeth over the toilet... So I totally get it! Talk to your DH and voice your concerns. You both need to be on a united front to get things moving (And honestly sometimes our DHs need a little shove to get them rolling)

Your expectations are totally reasonably for him.  That being said, I think that kids actually have brain damage at that age that makes them forget things and need to be reminded... repeatedly.  It must be frustrating because you certainly must be tired with a new baby and sure the hormones in your body probably don't help..lol.

I guess I might suggest to give yourself some mental fortification is that you can consider yourself a little lucky that the kids are generally being good about the baby.. wanting to touch/hold the baby etc..   It might make the fact that you must remind him a little easier.. perhaps he isn't intentionally doing this and is really scatter brained.

I guess a suggestion would to try to treat it as an inside joke with him about the washing up.  "Did you wash your hands... Really because we all know how much you like to skip that step.. Yeah.. I know you were hoping to save the germs for biology tomorrow but just humor me.. go wash them again.. please and thank you".