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Deadheading your Daylilies each day, will help the remainder flowers open and bloom fully.
When all the flowers have bloomed, you can then cut that stalk down to the plant base.
There are some varieties of Daylily that bloom more then once a growing season, but the vast majority will bloom only once.

Allow the foliage to die down naturally to feed the plant for next years flowers.

https://www.gardeningknowhow.com/ornamental/bulbs/daylily/growing-daylilies.htm

Cutting back the Bishop's Weeds should not effect the plants too terribly much, as they are quite hardy.
They may grow back slightly this season, but a recovering for next year is most likely.

https://www.gardeningknowhow.com/ornamental/groundcover/snow-on-the-mountain/snow-on-the-mountain-plant.htm

Spacing of plants should be 9 to 12 inches. Plant spread is approx 18".
You can generally order 'plugs' of the plants for planting in large areas.
Seeds can be directly sown into the soil but your coverage may not be even.
Follow seed sowing recommendations from the seed source.

Keep in mind that Snow On The Mountain can be invasive.
Here is a link with more information.

https://www.gardeningknowhow.com/ornamental/groundcover/snow-on-the-mountain/snow-on-the-mountain-plant.htm

This is no easy task, as you have clearly discovered!

This article may help with a few more suggestions.

http://savvygardening.com/goutweed/

These typically propagate by rhizome. the best way is to dig some of the roots up and plant portions of it. Dividing them will give you the best success with propagating these. They will grow roughly ankle-high. Here is an article with more information: https://www.gardeningknowhow.com/ornamental/groundcover/snow-on-the-mountain/snow-on-the-mountain-plant.htm

This looks like goutweed, aka bishops weed. Aegopodium podagraria. It spreads via white, fleshy underground rhizomes. To my knowledge, there is no "natural" remedy. You can burn it with a blow-torch device but it will come back. Re-treatment will be necessary and it may take several months or even years to be rid of it. Other options are to cover the area with thick black plastic and let the sun kill it (solarization) or cover it with an outdoor carpet. However, it may pop up just past the edge of your treatment area. If it is coming into your yard under a fence, you will not be entirely rid of it since it will creep back into your yard. To see it, do a search for "garden blow-torch".

The simple answer is that bishop's weed does not like hot weather. The browning is a partial dormancy and the plant will perk up after the weather cools. Keeping the soil moist helps it limp along during hot spells but it will not look great this time of year.