There are over 20,000 different varieties of rhododendron available, here's a few choice ones from an expert supplier local to me - https://www.rhododendrons.co.uk/rhododendrons/c9 We have quite a few ranging from dwarf to 5 metre tall and wide varieties.
You can cut back whatever overhangs your property boundary, and you can dig out roots that spread on to your side. Legally you're supposed to offer the cut material back to the owner (and dispose of it if they decline, don't just chuck it over the fence particularly if it's got seed heads on it). If it's council-owned land that it's growing on and it's blocking your access or causing other problems but not overhanging your boundary, they should deal with it. You could ask for permission to cut it back yourself (in writing/email just in case) if they can't or won't do it in a reasonable timescale.
Doncaster, South Yorkshire. Soil type: sandy, well-drained
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I live in west central Scotland - not where that photo is...