I think you are being a little overcautious - unless, perhaps, you know that you have a problem with foxes regularly visiting your garden, in which case I think it would be sensible to keep your little one close to the house. You don't need to worry about cats or birds.
I have always had pups out in the back garden from the day I get them home. They enjoy exploring the garden and running around, also they need to be taken out very frequently for toileting - on waking after a nap, after playing, after eating, after drinking, and whenever you see the body language cues of course.
It is an excellent idea to get used to taking your little one out and about with you, with the little one being carried of course. We bought a puppy sling from eBay for about £10, made it very easy and pup feels very safe cuddled up against you. Take her outside at the garden centre, up the high street, wherever you can - it should become a regular thing, meeting lots of new people and seeing as many new things as possible, all of which are just what you need for good socialisation.
I once tried to walk up Guildford High Street with our 10 week old Jess in the sling, tried being the operative word, so many people interested in Jess. There was a one man band as well, he came up to us and started on 'how much is that doggy in the window' complete with clashing cymbals.