Errr, not really, yes they had the name 'The Holy Roman Empire' but there was nothing Roman about them and they in no form, shape or way carried on the western half of the Roman Empire. Most of its territories where in Germany, where the Roman Empire at its height possessed little to no territory in.
It still included all of Italy and the Popes were the ones to call it "The Holy Roman Empire." It in a sense rebounded and expanded. But yeah I agree there are differences and it didn't become the Holy Roman Empire until a couple hundred years after the Western Roman Empire fell.
Yes. But the collapse of the Western Empire placed Western Europe into a dark age, And showed that the golden age of Rome will never come again since The Eastern Empire was slowly crumbling since then.
"The Dark Ages" weren't an age of total darkness as in no advances. When the Western Empire fell there was alot of crap, but things did rebound and science and technology advanced.