Please don't worry, it will get better!
I'm not quite in the same situation because I have dd whose just turned 2 and ds 4months. Dd doesn't lash out because up until a couple of weeks ago she was pretty much ignoring ds. We weren't particularly worried because we knew that as soon as ds became a little more interactive she'd soon start to take an interest. And she has. However, a lot of your anxieties ring bells with me in the early weeks! I'm by no means an expert but here are a few suggestions:
Keep as much to ds1's routine as possible. He will be far easier to manage if he feels that his life is still the same. Ds2 will fit in to that. I manipulated ds's feeds in the early days to be able to do this (never let him go hungry but sometimes anticipated him being hungry and so fed him slightly early if that fitted into dd's schedule better).
Don't worry about ds2 'missing out' on being on a playmat or anything like that. That's you putting your adult fears onto the situation. He's a newborn, being awake is stimulation enough! The fact that he's being breastfed and obviously cuddled by you is all he needs. Playmats are simply inventions by companies to get us to part with our money through guilt that our babies aren't having enough fun (of course I own a lovely playmat so I fell for it too).
Obviously watch out for ds1 being over enthusiastic with ds2 but remember that babies are pretty solid, despite seeming the opposite. Try to encourage ds1 to play gently around the baby and show him how gentle you are with him. Give him massive praise when he is gentle and try not to shout at him or over-react yourself (if you think, in hindsight, you might be!) unless he's actually being nasty to ds2. You do want to encourage ds1 to love his brother.
If you have a hands-on dh that's fab. Make sure the pair of you spend time with both boys individually rather than always as a family. And it only has to be 20mins kicking a ball around or 10mins singing songs to the baby. But that way you'll feel that both ds's are getting quality time with you. But ultimately imo it's your toddler you need to concentrate on - he understands that his world has changed and needs to be reassured that his place in it is still secure. The baby will be fine just tagging along!
And finally, ask for and accept all the help you can. I was a complete martyr with my first and stubbornly wanted to do it all myself feeling a failure if I didn't. With two under two I quickly realised that simply surviving each day was success in itself!
You sound like you're doing a fab job and if the only thing you're guilty of is getting cross with your toddler on 4 or 5 hours sleep a night, you've no claim on the bad mum crown! Getting out and about is the best policy and it sounds like far from being a failure you're actually coping really well. It is a hard slog but every day gets easier and more manageable. Particulary when you start getting more sleep.