I find this a really hard age. They are testing a lot and I am not naturally very good at boundaries.
The not being listened to is a huge trigger. I definitely feel like a shit parent and then it ends up as a self fulfilling prophecy, oops! I used to get this really awful powerless feeling like I couldn't make him do things I needed him to do and that kind of feeds into a panic and hit all the buttons hoping one of them is the emergency stop kind of reaction. It got better when he grew out of it, I'm afraid to say with that one. But it did get better.
My second one is three now so coming into this age as well, what's helped the most this time is that I have DH around who is really hands on - I was a single parent with DS1 and it was just me. In hindsight, I wasn't very good at being consistent with routines with him. DH is very good with routines and that helps a lot because if DS2 knows that bedtime always goes a certain way, he might complain a bit but he doesn't ultimately fight it. Whereas if we've been a bit slack with teeth brushing for example and then suddenly try to bring that in again all the delaying and resisting tactics come in and I could rage like some kind of dragon. And then once a few days of enforced teeth brushing have happened it's fine again.
The second thing that has helped is the how to talk app. I had read the how to talk book with DS1 and it really helped, but I'd forget everything when I was in that rage trigger state of "I just need you to listen to me". Now when that happens I can just break out my phone and quickly run through the tips (that are reminders from the book, you need to read the book first). This is a total game changer. I've used it several times with my 3yo and it gets us through the situation without me doing any shouting or crying 
It might not help in the moment, but another reframe that might be helpful is trying to remember that you cannot control your child's behaviour, indeed it is not your job to control your child's behaviour. The only thing you can control is her environment (to an extent) and your response to her. She is and always will be in control of herself. You can control things like what clothes are available to her, what transport is available to her, who she can see, what TV she is able to watch, etc. But she control things like whether she steps in a puddle, how much and whether she eats, which clothes out of the one she's given to put on, etc. You can guide her and influence her but she is ultimately her own person and that's ok. It's also ok for there to be consequences to her actions. As the adult, that's up to you to decide as part of the guiding and influencing part. That includes natural consequences such as her socks getting wet, or her shoes getting ruined - you can choose to override and protect her from those consequences by physically removing her from the puddle, or you can let her know you're worried about those things and then let her experience them if she really wants to. You might even decide that she needs to use some pocket money to replace the shoes if they are ruined (that is probably a bit much for a 5yo and this situation, but just as an example of how you can use your position as the adult to point out the consequences of something and build on them if necessary).
Worries like not wanting her to spoil her school shoes are valid, but it might be worth thinking whether her shoes are appropriate if they can not cope with a puddle. If she wants to puddle jump, maybe you can bring wellies to school or look at sturdier shoes for next winter.
I agree that it sounds like maybe you need your DH to step up more if you are feeling yourself get ragey with him as well or generally feeling like you are overwhelmed. With a 6mo as well if you are breastfeeding it can be well worth taking a multivitamin for pregnancy and breastfeeding. I was always quite anti any idea that breastfeeding could possibly be harmful but it certainly won't hurt and it might make a positive difference. If it doesn't help, just stop taking them.