Does the cat run need staking in any way so it doesn't topple over in high winds?

With the skirts attached, our runs are stable in winds. We always recommend using ground spikes or stakes if the run is on uneven ground. Staking is particularly important if you intend attached covers of any kind. In high winds we recommend that you remove any covers, as these will act like a sail. 

We do offer Screw Pegs designed to secure your run to the ground. Here's a link: https://www.omlet.co.uk/shop/chicken_keeping/chicken_keeping_spares/9361/omlet-screw-pegs-12-pack/.