You're going to get a lot of replies laughing at your question about relaxing with your child, but this has become easier for me since 2/2 and a half years old.
Here are answers for my 3 year old, but take this with a massive pinch of salt OP as all babies are different. My son was never the type of kid who threw himself about and had tantrums- that's just a temperament thing, not a parenting issue at your daughter's age. Some kids just do that and sone don't. He was, however, massively energetic and needed to be out all day long or I'd go insane. He's calmed down massively now, but we still spend most of the day outdoors.
So, for my 3 year old:
When did your little ones…
stop picking things up and putting them in their mouths? He still does this often now. I guess some kids are just orally fixated for longer. He's always got his fingers in his mouth too.
start eating independently with a spoon/fork? I don't recall. We did baby led weaning and he just ate what we ate. He's definitely been proficient with a fork and spoon for well over a year now
stop trying to get into everything? still curious. I still wouldn't take the baby gate off of our kitchen or leave him alone in the kitchen as nothing is toddler proofed in there and you never know!
stop throwing themselves back into a tantrum when you take something away from them (and any tips for dealing with this for a very young toddler?)? as I say, this is temperamental. Mine never did this. Just breezily move on and ignore tantrumming. But also make sure you're picking your battles. Tantrums might be fuelled by a sense of injustice. Otherwise, distract, ignore, and move on.
settle in the car and not throw a tantrum because they can’t stand up and look out the window? never had this issue. Let her have a minute to stand up and look out the windows, then stick a melty puff or something not too messy in one hand and a fidget type toy in the other and get her strapped in. Bright and breezy and just ignore it.
start to enjoy relaxing a bit more, eg just sitting or lying on the sofa with you? My baby is either asleep or crawling around/getting into everything and I’d love for us to enjoy resting a bit more together. Mumsnet will collectively piss themselves laughing at this one. Your baby is doing what babies must do if they have a certain temperament. Some babies are chill, some are curious and just want to be on the go. Mine was on the go really young and never stopped. But he's much more relaxed at 3 years old now.
We still go out most of the day as he thrives outdoors and it tires him out for bed at 6pm. But, he woke up at 6:40, and is now playing quietly in his room with a cup of milk and a couple of pancakes. He's reading his books and playing with his toys and I told him I'm not ready to be up yet so I'm relaxing in bed for a while. I'll get up at 7:30 probably. Eventually I hear him put a card in his Yoto player. He's very content, but most importantly he's at an age where I can explain and reason things with him. He knows to stay in his room quietly playing now until I'm up. We're also able to enjoy a cuddle on the couch and a bit of telly in the morning, or he'll play with his toys and leave me in peace to have a cup of tea on the couch. Movie nights with a pizza in front of a feature length kids film are possible now. Other posters will have children of the same age who cannot do these things, as with everything it's temperamental above all.
cope with going to restaurants? (We do go out to eat but it’s always extremely rushed and often results in us having to hold her) he's great in restaurants and cafes now but of course it wasn't like that at one year old. This is one that you have to put the work in to over time. Don't expect her to be settled for long in restaurants now. At that age you take everything you can to distract them, tag team taking them for little walks, and order only a main and get out asap. You just build this up over time. Even now if we're going to a cafe or restaurant we take a book and a sticker book with us, and I make sure he's had a really good run around outdoors before we can expect him to be settled for a couple of hours in a restaurant. Short cafe trip = book or sticker book, couple of hours in a restaurant=books, stickers, little fidget toys, small toy cars in a tin etc. As he gets older this will change I'm sure to just colouring in books or something like that.
seem a little less in danger of injuring themselves if you look away for 5 seconds? depends on the child. This started young for my son because he was up and about from a young age I think. He's just very physically able so whereas some of his little friends have had multiple A&E trips for injuries, he's never had more than a bruise here and there.
Seem to enjoy hanging out with the family rather than being dragged along? I think this got easier for us when his language got to the point where we could reason more and have little conversations, so from the age of two onwards. He's very much a little friend now but of course at 3 we still have moments. Toddlers are often quite selfish by nature. They learn to come and go with other people over time. Again, it's easier if they have a more easy going temperament and/or the language skills to talk things through.
It all takes time and hard work OP. I'm really big on putting the hard work in now for an easier time in the future. Playing the long game in the early days to get mornings like this one now where my son is still playing in his room nearly an hour later, and I'm only now considering getting up for the day.