Have seen many vanities butted up against walls on all 3 sides and 2 sides as well as the usual wall below a mirror. No idea of operation of drawer/etc. until one was seen in the flesh. Depends on hinges, etc., what its being installed near, etc. Basin top taps are probably easiest to install. They also generally use a stop cock to turn water on or off (for maintenance & the like) underneath the vanity with the basin top taps. More aesthetics than anything else really.

A lot of chaise lounges come with the ability to be switched either side at your whim. You could try and see what best fits. If yours is fixed, my vote is against the window. But you could mark out on floor with masking tape and see which you prefer.

Option 1 - having the single chair on the left makes the room appear smaller. Option 2 - will have a more open feeling when walking up the stairs. Option 3 - you could have the chaise under the window on the right side with the single chair and side table backing on to the dining area. This way the living area would be easier to access and should give a better sense of space. However this arrangement would not be ideal when viewing the TV. The entertainment unit could be moved a bit further to the left. Why don't you try Option 1 for a week or two and if you don't like it, change to Option 2 or 3?

It's a little unconventional badobbs, but have you considered a toilet with in-built basin and tap - these are very sustainable (although not for everyone!) and are perfect for tight spaces. https://specify.caroma.com.au/products/toilet-suites/profile/profile-5-toilet-suite-deluxe-with-integrated-hand-basin.  To answer your question, usually you need min 800-900mm or so for a fixed panel for a shower, however normally walk-in open showers are located perpendicular to a room, so that splashes and wet floors are contained to one end. Think of when you drop the shower head and it sprays water everywhere! In your case I would advise selecting a well made frameless hinged door - so you can still have a walk-in effect but a bit more protection from water spraying out everywhere. Try Reece; https://www.reece.com.au/assets/brands/244000/custom-glass-brochure-v1.pdf The vanity depth is going to be the issue here, usually we advise min 800-850mm clearance (although can go down to 700 as per Dr Retro House Calls) and can see from the plan that with an 800 bath and 700 clearance you only have 250 depth for your vanity when it's near the bath. You could get a vanity made up at 250mm deep with a semi-inset deeper basin centrally - that way you can make the vanity a little longer but with a workable basin. https://specify.caroma.com.au/products/basins/luna/contemporary-luna-semi-recessed-basin-without-tap-landing Good Luck!