So, what you are calling 'ground ivy' looks like what I call vinca.
Does that look like a match to you?
It is probably growing there against the shed because that is a dark, dry area which is difficult to grow anything else in. There are some alternatives, but you might want to leave it there for now while you tackle the rest of the garden. Rip it out where it starts getting into other plants.
When you say 'rotted leaves' do you mean dead ones, literally rotting on the ground? Leave those. They are free fertilizer. Or do you mean dried up ones on a plant? Take a pair of shears and snip off the spent blooms if they are unsightly to you.
It shouldn't harm bulbs if they accidentally get uprooted while weeding. Just tuck them back in, pointy side up.
It might help to identify what bulbs you have. One that is a common weed is
Star of Bethlehem (Ornithogalum umbellatum) They do have flowers, but not very many compared to the amount of leaves. You can keep them or toss them as you please.
You last picture looks like it has grass and
Creeping Charlie (Glechoma hederacea) invading the dead nettle. This is annoying. Creeping charlie is hard to get rid of when it starts mixing with other things. You pretty much have to get down on your hands and knees and trace along the stems and pull them out one by one. Repeatedly as they re-grow all summer.
You can pull the grass by hand, or use a selective herbicide like Sethoxydim which will (slowly, over the course of the summer) kill the grass and leave the ground cover.
Watch the nettle and think about how it looks through the year, and whether you want to keep it long-term. I would leave it for now as better than bare dirt, but if you find it messy, you may want to scout out something else for next year.
You have a couple of what look like young trees coming out of the nettle as well. Either dig them out, or snip them off and immediately apply glyphosate (roundup) to the cut end with a paintbrush to keep them from resprouting.
Get some woodchip or bark mulch of your choice and apply to the bare dirt. It will add organic matter, keep the roots cool, and suppress weed seedlings. I forget what official guidelines for mulch depth are, but I usually aim for about 2-3 inches.
Keep things watered, and see what else you have this summer!
(n.b. you used the term 'annual garden' in your post title. All the plants pictured with the exception of your marigolds are perennials. Annuals grow from seed and die each winter. Perennials survive year-to-year)