No to your first question. Even if we restrict ourselves to anatomically modern humans, the earliest such inhabitants of the British Isles (at least 4000 B.C.) predate the branching off of Celtic from other Indo-European languages (proto-Celtic is generally dated to around 800 B.C.). There were certainly non-Celtic Romans who conquered Britain in Classical times, and there is some dispute as to whether even the pre-Roman inhabitants of Britain should be called Celts (although this is mostly a matter of definition as they probably spoke Celtic languages). The Picts are generally assumed to have spoken a Celtic language, but there is not enough evidence.

The Normans are originally Norse people who settled in Normandy in Northern France and adopted a Gallo-Romance language similar to Central French. England came under the rule of William of Normandy, but the Normans have little to do with the Scots.

Today's inhabitants of Britain—the English, Welsh, Scots—are a mix of all those historical peoples that passed through the isles, including the Celts, Romans, Anglo-Saxons, Danes, and Normans. The traditional language of the highland Scots (Scottish Gaelic) is a Celtic tongue. The Bretons in France also traditionally speak a Celtic language, although this is now moribund. Celtic languages are extinct elsewhere, but vast numbers of Europeans from Iberia to the Balkans can count Celts as their ancestors.

No, there were most certainly non-Celtic peoples inhabiting Insular Europe before the arrival of Celtic languages and material culture ~500 BC, though we don't really know what kind of languages they spoke or how their societies were structured.

It's possible that non-Indo-European speakers survived in Ireland until just before the introduction of Christianity there based on linguistic evidence, and genealogical and historical records (despite being fundamentally flawed sources) suggest migrations by Continental Celtic speakers from Gaul or Belgium in pre-historic times; for example, the Laigin who gave their name to Leinster appear to have Gallic or possibly British origins. The names of some other archaic populations groups also suggest continental origins: the Bibraige ('people of the Beaver') have as their totemic animal a species that has never been native to Ireland, meaning that they likely migrated from Continental Europe. As well, possibly-Gaelicized populations of Britons/Picts seem to have existed in the north of Ireland until about the 9th century AD, not to mention the establishment of Irish colonies in Wales, Cornwall and Scotland (in fact, that Irish colony would become Scotland!) following the withdrawal of the Roman Empire from Britain. Population movement was much more common in early Irish history than many seem to think; different cultures and populations were constantly interacting with one-another, getting involved in conflicts with one another, displacing each other's aristocracies etc.