Wow, I don't think I've ever seen powdery mildew on citrus, but this sure looks like what you're dealing with. First, I would do some careful pruning to open up the canopy and improve air movement through the tree. Any branches touching the ground should be pruned at the trunk. And, I would use Serenade Spray, which is organic, and will not harm your tree (or you). I would also make sure your tree receives as much sunlight as possible and make sure the sprinklers are not hitting the tree, but the tree is receiving water only at the base of the tree at the drip line.

Patty S.

Here is a link that might be useful: Master Gardeners Santa Clara County - Powdery Mildew

I have pruned the tree and after a few days of shower sunlight is showing its way through :)

Might be a stupid question --> Could we eat the lemons (lemon juice/pickle) that are currently growing on this tree?

The white stuff isn't a disease.
But is evidence of whiteflies. Most likely woolly whitefly.

If you didn't apply a pesticide, yes, you can eat the fruit, even today.
*But* if you did spray, you must observe the waiting period ("pre-harvest interval" or PHI) stated on the label.