I can't ID it from the photo but the general rule of thumb when pruning flowering shrubs is to do it immediately after flowering. One good reason for this is that many have their flower buds on the tips of last year's growth and those will be lost or won't have time to develop properly if pruned at other times.
Yep, asparagus for me too.
They're close relations but I think you are right Asarum as buttonwoods are rare in the UK.
Platanus occidentalis (American plane) aka Buttonwood, I think.
Also adding a few heucheras will give year round colour. Many are shade tolerant:
They tend to do this if left alone but respond very well to pruning. They can be cut back as hard as is necessary and will quickly form new shoots.
There are some 'long season' mixes around which are worth trying - for example, this one contains 3 varieties:
Hopefully, at least one of them will grow well for you and if all 3 do you will have an extended season without having to successionally sow (which doesn't always work well depending on the weather.)
Reminds me of something - Calendula?
It does sound like vine weevil. If you use the bay leaves in cooking, I would avoid any kind of insecticide. That really only leaves vine weevil nematodes which you apply to the soil. These kill the grubs in the soil (which eat roots.) The adults are mainly active at night so you may be able to catch them on the leaves by doing moonlight visits with a torch.
I would plant them out in their final places now. If you have a greenhouse then you could pot them up and place in there which will bring them on earlier but they won't do well grown inside a house as it is too warm - they are fully hardy and will survive frost once planted.