if you want to be able to open the doors, load the boot and walk around i think you need at least 9m x 7m but if you to also want useable work benches and storage spaces (eg for the lawn mower etc) add at least 1 m to width and length

Can you post a sketch? I'm scrolling back and forth trying to visualise it all and understand dimensions and orientation and usage. If the fence colour suits the house and creates a good background for the plants I wouldn't worry about whether it will date. The same concern could apply to the house, but if it all comes together in a way that is classically timeless rather than overly trendy and you're happy with it, then it's working. Alternatively formal or informal hedging can obscure the fence and avoid the question of what colour the fence should be. I agree that with a toddler and dogs you want to keep as much of that lawn as you can. Work out how much garden bed is needed around the edge so that you are not constantly pruning the plants back to fit, but so it still leaves as much lawn as you want. What's missing in my view is edging, most garden styles look better with edging between the plants and the grass. Another thing that gives gardens a more finished look is underplanting. Research plants to ensure you don't plant anything that is toxic or allergenic. Remember humans and dogs can differ in that regard. Start an Ideabook for your garden. If you have images of what you are trying to achieve, and you can share that with us, you can make progress with specific how to's and choosing specific plants.

Please provide some suggestions on how can I fix this slope. Is retaining wall the only solution or can something easier and cheaper be done? Thank you.