The others have said it all really
Start again letting her settle then do some bathtub taming as you won't be able to handle her until she's tamed. You started out well leaving her alone for a few days, but she isn't tame yet and yes they will wriggle free, ping and try and escape and run. This is a tricky time for getting them out of the cage to do some taming. It's best to put them into something secure like a hamster ball - to transport them to the place you can let them out for taming (dry bathtub is good).
Either by letting them walk into a tube with a treat at the end then lifting the tube out with your hands over the end and then, with the ball nearby, putting one end of the tube into the ball and keep your hand over the other end. So she walks into the ball then pop the lid on. Or a large box will do but it needs to be high enough she can't jump out. So then you just put the tube down in the box.
Then carry the ball with her in (careful you don't drop it) to the bathtub put the ball down, take the lid off and let her walk out. A few toys and hidey places in there can help - eg a cardboard tube, a mug on its side another two or two. They tend to just try and get out and slither up and down the sides of the bath but she might investigate the toys. During this time you can get her used to touch. Once they learn to trust your touch they are handleable without "pinging" (that is the wriggling free and leaping 2 feet in the air at the same time).
Tips I had that worked were you start by just stroking her on the back with one finger while she's in the bath tub - quite easy to do if she's slithering up and down the side of the bath, but wear gloves if you're nervous she might bite. Our hamster used to jerk his head round to see you off/threaten to bite but never actually did. Do this a couple of times in the first session but just let her explore mostly and talk to her. They usually get back in the ball when they've had enough and then you can pop the lid on, lower the ball in her cage and let her out again.
Maybe do bathtub taming every 2 or 3 days. By the second or third time she should accept being stroked on the back with one finger without jerking round or biting. This is real progress! She is accepting touch.
From there you move onto putting your hand flat on the bathtub, palm up, in front of her so she walks across your hand. Do this for one session. Next session do that again but very slightly lift your hand about 1cm - but don't try and hold her - just let her walk off your raised hand. Maybe do that 2 or 3 times. Next session do the same again but raise your hand a bit higher - maybe an inch or two - but still let her walk off. Do that a few times. Then let her walk from hand to hand putting one hand in front of the other - still palm up. Some people let them walk through your tunnelled hands at this stage but I sometimes find they don't like that and pull back.
Once she is walking across your hands you can hold her and pet her. But only try and pick her up and hold her for a few seconds at first, in the bathtub. But she should be more relaxed and feel trusting with you holding her.
But before you do all the taming I would literally start from scratch again now. Leave her alone for 2 or 3 days then don't clean anything at all for 2 weeks. I think 2 weeks does it. No spot cleaning, no wheel cleaning. It'll be fine! Let her find her feet, get some habits, make the place her own.
The only thing that smells is their pee. It might start to get a bit whiffy. The earliest I would spot clean would be after one week but only the main pee area -eg take out a handful of the pee'd on substrate and replace it with a clean handful. In your case, if you can bear it, I wouldn't even do that and leave her a full two weeks.
As a tip, after that you will know where her chosen pee area is - often a corner of the cage - and you can then put a corner litter tray there with chinchilla sand in and she should use it. Syrians will use a toilet. Then you just need to empty the litter tray once or twice a week and the rest of the cage should stay clean and dry.
During that two weeks just talk to her through the cage so she gets familiar with your voice and presence. Tone of voice makes a big difference. I would start with saying you are very sorry for having to grab her but you were trying to protect her from falling and hurting herself and it won't happen again. I know that sounds a bit bonkers but when you say sorry you have a tone of voice that sounds sorry! And they listen and it does make a difference. Otherwise just chat to her in a friendly tone and tell her she's clever and beautiful etc.
After a few days you could start offering her treats through the mesh (I assume the bin cage is meshed?) So she gets used to taking things from you.
If she shows signs of wanting to come out during that two weeks then its fine to start with bathtub taming. But if she doesn't I'd just leave her for now.
One thing that's tricky with a scared hamster is seeing a hand coming from above - they are prey animals. So to get her out, lowering a tube and letting her walk in is better (or lowering the hamster ball). Some hamsters will get used to that when they're tame but it's one reason I prefer cages so you have a front entrance and no hand coming from above.
She is still a baby! So this is all fear. She is scared - she had a fright when you had to hold onto her, she is scared that someone is invading her cage and messing with it. And this hand keeps coming from above and invading her space!
They really need that two weeks to make it their own - scent marking makes it there own and you remove their scent when you clean. Scent marking also helps them find their way around - they don't see well so they follow their scent trails.
I've had hamsters refuse to use a wheel after it's been cleaned because all their scent has been removed. Try not to worry about the wheel to much just yet - most hamsters pee in their wheel - it doesn't need a full wash every week - a quick wipe inside now and then (but not every week) is fine for a longish period of time - but right now just leave it for two weeks. The pee will dry and it'll smell less!
It can become a catch 22 as well in that the more you clean, the more they pee on things to deter someone from messing with them!
It is very early days and they are still babies/children for the first few months
They get into better habits as they mature a bit.
One tip though -their nest and hoard are their most precious things so avoid spot cleaning those - you can go a very long time leaving a nest and hoard in tact. Unless they get pee'd on (baby hamsters do sometimes pee in the nest but usually grow out of that quite quickly). She will see you as a thief if you mess with her nest and hoard! For now - just leave them - even if pee'd in.
But soothe her with talking to her through the cage for now. They forgive you however much you think you have had a disaster - but they just need to calm down and feel secure and in control of their space until she's fully established and feels at home.
Some people also recommend doing the tissue trick to help them familiarise with your scent. You put a few sheets of toilet paper up your sleeve or down your bra and leave it there for a couple of hours. Then tear the sheets into strips and add the pile of paper strips to the cage - not in her house, just somewhere in the cage. Add them while she's asleep. She will pouch them/forage for them to take to her nest and will become familiar with your scent from them.
What are the dimensions of the bin cage? Some of them slope in a lot at the bottom and don't have a lot of floor space for a syrian. If it's a bit on the small side she may have a bit of cage rage as well. But it sounds mainly about removing things and cleaning her too soon.
After two weeks I spot clean the pee corner and that's it. And add a litter tray/empty it every few days. Maybe wipe the wheel after 3 weeks. You can easily go three months without doing a full cage clean, if you have at least 4" deep substrate and a litter tray, and even then it's better to do "partial" cleans - ie don't clean everything at the same time - which is our human instinct - it's better to always leave something smelling familiar. So if you change the substrate, don't clean anything else and keep back some of the old substrate that's clean and spread it on top of the new - it makes a huge difference to avoiding them being stressed. That top layer will get spot cleaned out eventually.
Then another week do the wheel. Another week again any toys (toys often don't need cleaning very often). Then another week check inside her house. If it's pee'd in then you'll need to remove the nest and any soggy hoard but always leave a bit behind. A bit of the old nest and a bit of the old hoard, that is dry - even if whiffy and always replace the hoard with some new food in the same place. And put more piles of paper strips out in the cage so she can rebuild the nest. They soon learn not to pee in the nest if it needs removing.