If things are cold, wet, and slow, I think I would hold off any heavy mulching for now.

If your soil is wet and cold now is not a good time to mulch since that can keep that soil wet and cold much longer. Mulches can keep many plants, wanted or unwanted, from growing if the mulch is applied thick enough, but there are also some plants that will burst through mulch with no problem. Daffodils, Tulips, Hyacinths, Crocus, etc. are pushing their way through the mulches on my planting beds now although some have yellowish tinges on the leaves coming through. That may be due to lack of sunlight or maybe a Phosphorus problem due to cold soil.
I would wait until the soil is a bit warmer and perhaps dryer before applying any mulch.

Is it hard to mulch when the perennials have emerged because they fill the bed? If so, there shouldn't be a huge need to mulch to control weeds - the perennials themselves will discourage them. If you are mulching to replenish organic matter and add to soil fertility, then I'd elect to do that in late fall, after you've cleaned up the area for winter.

Yes, mulch can potentially slow or retard the development of any plant depending on size. Certainly it will do so with annuals or any self-seeding perennials but it also can smother other, less robust types of herbaceous perennials before they emerge from the ground. It is also just as likely to promote fungal issues and crown rots if spread completely over perennials.