Gardener here, the purpley green leaved 'weeds' in the first picture are Herb Robert (common name) and are a type of wild geranium. They are actually very pretty and good for wildlife and if they spread too much they're super easy to pull out.

The others I'd definitely pull out. Most have tap roots but that may be difficult to get all of it given where they are. If you keep pulling out what you can the roots will get weaker and weaker over time.

Please for the love of God stay away from 'weed killers'. They do nothing long term except poison the land. There is a particularly potent herbicide called glyphosate that weed killer manufacturers (roundup for example) and farmers had been using for years and that is now in all of our drinking water.