Hello Lynda.
I am pretty new to this chicken-keeping malarkey too, having only had my girls for just over a month but, for what it's worth, I can give you my thoughts until someone more experienced comes along...
I have four birds but a nesting box for two - it is not where they sleep (well, they're not supposed to but mine are ex-batts and, hey, if they want to sleep in the nest box then I'm not bothered) but just where - in theory - they go to do egg things so you don't need the same number of nest boxes as you do hens.
I have learned from this site that two hens is a bad thing - if something happens to one you are left with only (lonely) hen - so I'd stick to your plan for three.
Hens can jump, apparently over six foot fences if they put their minds to it. When you say "the fence around" then I assume you are talking about the fencing around the coop and the run that came with it? If so, then 3ft may or may not keep them in, depending on how badly they want to get out. If you don't mind them doing a Great Escape bid into the rest of the garden then it'll be fine. If you want that little fence to keep them in, I'm not sure it will!
Your garden fence where it's only 4ft needs to be higher - to keep the chooks in but also the dog out! If the dog spots the chooks, it'll be over that fence pretty quick. The girls might fly over but hearing the dog might be enough to keep them away. I'd be more worried about the dog jumping over.
You can clip their wings easily enough yourself. See here:
http://www.omlet.co.uk/guide/guide.php?view=Chickens&cat=Chicken%20Care&sub=wing%20clippingThat's my humble and inexperienced offering! Good luck and enjoy your girls!