It is just hard and yes it is normal. 2 yr olds are great with selective hearing! Good news is eventually it will get better. Distraction and thinking ahead is really key. You can avoid lots of tantrums by giving warning in advance, eg in 5 mins we are putting the toys away, do you remember in 2 mins we will tidy up etc! Or turn things into races, or just be plain daft. For example. Refusal to put on coat might lead to Mummy deciding it is her coat and trying to put it on in a comedy fashion. Usually it will be ripped out of your hands within seconds after that. Also, never let him get tired or hungry. Pre-empt low blood sugar with bananas, cubes of cheese etc at 2-3hr intervals.
If he goes for his sister, distract, distract, distract. Maybe even have some pre-prepared rouses (wow, look an aeroplane/squirrel/spider/unicorn!!). Distraction for 2 year olds doesn't have to be high brow, or even make sense, in order to be fun e.g."can you wrap yourself up in this toilet roll?", "Look at these bright post-it notes, can you put one under every chair in the house?". "Wow, I bet you can't find all the notes again can you?". 2 yr olds love repetition grits teeth.
Can you try to phrase the negatives positively? (eg, if toddler were to move to bite baby, intercept and say "gentle kisses, remember"; or if they too enthusiastically pat them on the head or pull hair, guide their hand and say "stroke pandora's hair gently like this, that's nice" with a big smile grimace". I always felt a tit, and it was hard to stay calm, especially when sleep deprived, but they do eventually get the idea. Admittedly, not the first time, you have to persist. Physically redirecting them gently before or during the act is more effective than telling them off afterwards (though totally understandable!). Very few people find all this easy.
Toddlers are like Labradors -fresh air and exercise twice a day 
. Far less interesting to prod baby if you're in a playground, though of course that is harder in the current weather. A waterproof snowsuit is a godsend.
I found baby on floor with young toddler in proximity didn't really work. It is harder work as they start to crawl then cruise and need to be able to stretch out. You need to be down there with them and present all the time with eyes in the back of your head more than when the baby was just carried around. Some people use a playpen or room divider to keep baby separated from toddler for times when you can't do that.
Good luck, and remember you're doing a really good job. You don't have to be perfect all the time, we all shout sometimes a lot. Just give them a hug and a kiss afterwards and say sorry. Crying is OK too, we all have bad days. Have a (((hug))).