Deck past it's prime- well, before ripping out and starting over, can you try sanding it? Home Depot and Lowes often rent those big floor sanders and for a deck you certainly don't need to be an "expert" (floor sanders can cause uneven sanding if you don't do it right, but on a deck this wouldn't be as much of an issue as it would in a living room). Or a small deck you could even do with a hand belt sander (though it would take some time and not exactly fall into the "fun" category for me). You could also just do the edges and cover the center with an outdoor rug or other covering, or cover the whole thing with something more foot friendly (you know those bamboo blinds I mentioned? I put an extra down next to the girl's outdoor gravel/sand pit as a sort of ground cover... some plants have grown through it, but it's a nice compromise between hardscape and nothing).
Sun and bugs- well, there's the classic deck set with an umbrella. Or there are sun sails (fabric triangles that attach to the house and a freestanding pole) which you can buy or make. If you have the wood and craftiness you could build an arbor (attached to the house or free standing) and plant climbing vines on it or drape fabric through it to create shade and interest/privacy. If you have more money than time though you can go to a garden supply store, HD or Lowes, or even Big Lots and find those freestanding plant arbors (some are flat, traditionally put against the wall of a house for plants to climb, and some are actual arched arbors that are only a few feet "deep" designed to be an entrance to a garden or to cover a bench). You could line one or more sides of the deck with these and use them to anchor a ceiling of fabric, or to create privacy walls, or mark out space without affecting the open feel too much.
And carbon dioxide traps are pretty effective at bug trapping without being messy/dangerous to kiddos and pets.
furniture- do you have a hand drill? Or a hammer/nail? Poke holes through the seat of the chair or wherever the water gathers and the rain should just drain right through. My MIL stores cushions and other things that might stay squishy or get funky when wet in a tupperware type box under the patio table as well and then tosses the cushions onto chairs as necessary.
Crafts- we dug a hole, lined it with landscape fabric, edged it with split logs from a tree we took down, and filled it with sand and gravel. The girls love digging in it, the rain drains out quickly, and the bugs don't come up through the fabric. Plus the gravel discourages cats from making it their litter box.
They also have a water play table. I don't really clean out the water table much... if I notice it's getting really nasty I may dump it, spray it with a little bleach water, or just leave it turned to the sun for sun bleaching. If it rains I sort of feel like it's a bonus since it fills on it's own! For actual "messy art play" maybe a few flowerpots (they drain on their own) to hold supplies and a few of the bigger flowerpots turned over to make fun kid sized seats? I guess it depends on what sort of art they like to do. Maybe water sensitive items could be kept inside by the door or in a waterproof tub by the play area?