Get a semi-decent looking storage "tower" - there are some with wood frames and fabric "drawers", or metal frames with wicker baskets, etc. Sometimes they have little labels or a space for them - you can use a photo for what goes in each one. For example, Legos in one, animals in another, crayons/pads in another, etc. When it's time to clean up, you remove the bins and have the photo (from the box it came in, or one you took, or something from the internet) facing her. She plays "the matching game" and tries to put things in the appropriate box. Then you carry the 3 or 4 bins to the shelf unit and put them in, in any order, photo facing out. Make clean up part of the game. Most preschools and kindergartens have little "clean up time" songs they sing while doing this.
I saw a cute room divider/decor item on TV - buy one of those folding screens at a discount store, yard sale, consignment shop. It doesn't matter how yucky the screen part is. Then get a roll of wallpaper - self-stick - and cut plain sheets the size of the panels, and stick them in. The screen can be in the corner of the living room or the dining room, and any big stuff can go behind it - instant clean-up. That works for the baby equipment you aren't using but want to have handy.
Put the plastic storage bins in the basement - rotate her toys upstairs in manageable amounts. She can't possibly play with everything at once if you don't make it possible. Label the bins with the photos the same way you do the "drawers" above. That way you know what's where.
Can you get a decorative hamper at a yard sale? It seems that it is taking up valuable space in the closet. Not sure where your washing machine is - on the main floor or in the basement. If it's in the basement, can you put a hanging laundry bag on the inside of the basement door, where you can just stash stuff and grab it when going downstairs? Take the laundry down, throw it in the washer, bring up a plastic bin of toys on the way up?
Use a see-through plastic shoe hanging bag on any available closet doors or the basement door - use that for stashing your 5 year old's smaller toys at her level, and stuff you need at the upper levels (microfiber duster or rags, extra tubes of diaper cream, burp cloths, anything like that for the baby. If the pockets are see-through, it's easier for your daughter (and you) to see what's there, but again, you can use a photo for your daughter and, now that she's learning to read, put a written label up there as well (don't rely just on written tags though - pre-readers and many older kids are still very visual vs. "literate").
Stop worrying about people dropping over! You have a 3 week old baby and a 5 year old. Get a sign for your front hall that says "My house was clean yesterday, sorry you missed it." I have one, and my "baby" is out of college! Embrace the chaos, and repeat after me: "Dull women have immaculate homes, dull women have immaculate homes."