Wow! 2 years ago, I could have written that exact post :-). Mine are 22 months apart and I was almost 30 when the second was born.
Yes, it is hard to enjoy the moments, especially when you're sleep-deprived. You're not a bad mom, you're just a tired woman. I fully believe that the sleep-deprivation is the key to most of the stress/short-temper when you have 2 little ones like this. I promise there's light at the end of the tunnel. When the baby crosses that 8-hr mark at night, you'll feel like a new woman all of a sudden. In fact, see if there's a way you can take a night off here and there. Maybe your husband can take baby duty, maybe a grandma can come sleep in the baby's room one night and take over for a little while. Or maybe the 2-yr-old can spend a couple days with an aunt or grandma. We did that every couple months after our second child was born and man, were those breaks worth it. If you have family or close freinds near-by, it's ok to lean on them right now! Besides the grandma help, I have a couple close friends who would come by occasionally just to take my older child to the park for a little while.
You're definitely doing a good thing by getting up a little earlier. Even if I manage some alone time during afternoon naps, it's not the same as STARTING my day with a few minutes to just breathe, and spend time with God. There is no way to make it through parenting without regular communication with God. In about 6 months, when you have had some sleep and can read a book...I'm finding some inspiration in the book Sacred Parenting by Gary Thomas.
And it's GREAT that you can spend a few minutes with the 2 yr old. The baby is taking more time away from him than you probably realize, so figuring out a couple of little 15-20 minute intervals where he gets you all to himself is a great idea. It helps to remember that right now, the baby won't remember if she had to wait 15 minutes for a meal or a diaper change, but your son will remember and react later on to missing those 15 minutes. However, your toddler is not going to be scarred for life if you have more bad days than good right now. I often feel so guilty about that first year of my second child's life. I feel like I didn't give my firstborn the time and attention she needed, etc. But you know what? She doesn't know or remember a thing about it. Did she watch a TON of TV/DVDs that year (at least the first half of it)? You bet -- how else would I be able to feed the baby (or myself for that matter) or take a shower? Is she any worse for wear? No.
Your house will never be as clean as you want it to be right now. If you can afford it, hire help. If you can't, find a closet that you don't need to open very often and stuff things in it :-). I found it helps to use baskets. I filled book shelves in the living areas and kids rooms with baskets to make toy pick-up easier. Even though pieces and parts of things were scattered among different baskets, at least they weren't scattered on the floor. Then I got a big canvas laundry basket to keep in our bedroom. All the junk that I didn't have time to sort through got piled in there.
Also, one day I just had to ask God to help me let go of my frustration at the state of my house, and it worked. I still have to go back to that prayer, but somehow God has helped me refrain from having my skin crawl at the disorder.
I also realized that there is a direct trade-off between the cleanliness of my house and the amount of quality time I get with my kids. Even now, if my house is clean, the kids are likely grumpy and begging for attention.
2-yr-olds are their own special kind of creature and take tons of energy to deal with. Now that my oldest is 4 and the youngest is 2, I wonder how I even survived that year with a 2 yr old and a newborn! At least this time around, the other child in the house can get herself a drink and snack, and use the toilet without my assistance!
He's going to make messes, try to take control of any situation, and throw a tantrum at the drop of a hat. Do your best to give him some independence when it's appropriate, and make good use of a playpen or baby gate when he just needs a break (or you need a break from him!).
And, no, I did NOT try to get my 2 yr old to clean up her own mess, pick up toys, etc. It took too much energy to try to train/coordinate that. I did my own quick pick-up while she watched a video or napped. I waited until she was 3 to start on that and now, at 4, she can clean her room and make her bed independently.
As soon as the baby was old enough to really react and find pleasure in anyone but mommy, it was delightful to see the two kids together. No one could make my son smile and laugh like his big sister could. He watched every move she made and to this day things she hung the moon (yes, he even lets her dress him in tu-tus and runs around playing "princess" -- much to his father's dismay).
Anyway, I know this is a hodge-podge of advice, but I hope it helps.