Hi Urbanite, welcome to G O Y , you can certainly cut your hedge back, go as low as you like, it will green up again in about 4 to 6 weeks, cut a couple of inches below the height you want, so you can trim the hedge without hitting thicker wood, just the new growth.
I don'tknow of any reason not to plant anything you want where privets have been, and you can shred what you cut off, with no danger of any privet growing back, I hope this helps you, Derek.

Totally agree with Derek... You may not be able to get all the stumps out in which case use SBK on them.

I think when you finally remove the privet for your new flower bed the soil will be very poor as the privet will have taken the goodness out of it.
So it will need enriching before planting such as good compost/topsoil

I agree with Pamg
Ref the soil being of poor quality as a lot of people never feed hedges and when they plant the like's of gladiolus
near to the bottom of hedges they dont take to well because the soil is very poor near the hedge and by the second year the bulb hasn't had enough feed to flower well;

Now if you want to grow verios plants once you've removed the hedge!
As well as doing something ref improving the soil "thats one job to think about as hydrangea 'zebra' needs a good fertile soil and regular watering when first planted and because of thin rooting system a good mulch during the winter.
But make sure the shrubs your going to plant there are of the correct type of soil your replacing ie

The zebra is a lovely greenish later going to white heads,

I can see from your photo's how you've not got a lot of natural light in your lounge,
Good idea to remove the hedge 'but it may well show a nice green leaved on both sides "but the top may stay without leaves for the summer,.

Once you have it down to about three feet and its greened up again you might even find you prefer it to railings!
I've found privet quite possible to dig out as mine seemed to be quite shallow rooted and if you miss bits it doesn't seem to grow up again from what's left. PamG's right though, if you do take it out you'll need lots of humus added to the soil as privet is very greedy.

Thanks for the all advice. I really don't think I will ever chose privet over a mixed border and this is 50% of the garden space (a similar size border in the back yard) everything else will be in pots.