There will be a height difference, you will need to use a reducing strip into the study and lounge. Don't forget the timber floors will also reduce the height of the skirts unless you are planning to take them up. Will you be continuing the flooring under the dishwasher & fridge, this will also change their heights. In the long run better to take up the floor, there are companies that will do it in a day, leave the floors prepared and take away the rubbish for around $30 m2.

Would be good to use as a sitting area for BBQ's etc, rather than blocking it off with screening. Probably not ideal for storage though, as it looks like it gets damp/wet. I've done one recently (surprisingly similar in many ways) where we are to excavate down to gain extra head height. We will then lay a new slab and use screening to the sides facing the neighbours. We are to also tile the deck above and waterproof it, etc.. It will add quite a bit of usable space to the whole site and house. It'll be used for BBQ's, etc., cheap way to increase the entertainment areas of both thats for sure! You could do something similar depending on budget. Better to update the area and then use it rather than just blocking it off as it will become wasted but valuable space.

I am installing MSI Everlife LVT waterproof vinyl flooring. I understand I can't use rubber-backed rugs. Is there a pad I could put under the rugs?

At first I thought OMG what are you thinking with putting a grey kitchen in with all that beautiful wood. But I did a google search, and found the colours look good together. I would keep the floor a neutral colour and pattern. Not concrete look, not wood look, definitely not slate look. Refer to the photo below, a neutral floor with the grey kitchen, let the wood be the feature.