What might be the impact on Britain if
Vortigern had not brought the Saxons to Britain as mercenaries?
Firstly the language, instead of being closely related to Frisian would be more Scandinavian or Germanic I'd imagine, assuming the Normans effectively assumed the role of the Saxons. Otherwise perhaps we'd still have widespread celtic languages, albeit influenced by Norse and Viking culture.
How does this affect the Britons battle against Picts and Hibernians?
Not a huge amount imo, I'd imagine left to their own devices they continue to fight against each other and themselves for eternity. The borders may be redrawn from time to time but I'd guesstimate the power balance between gaels and britons to be fairly even. Whatever role the picts play is uncertain but they didn't do much to halt the invasion of the gaels.
Is it possible for Romanized Britain to survive?
Not in that form, there are endless threats to the British Isles, the Saxons lessened those threats by uniting England but without that they lie even more vulnerable.
Is Britain doomed to splinter into dozens of states or can it remain unified?
Ireland was divided into five states, Scotland is debateable because the emergence of Scotland as a nation was influenced by the existence (and threat) of England. I'd say that except the brief emergence of Mann and Dalriata etc, generally the regions would be divided into the five provinces in Ireland, The Highlands and Islands of Scotland, an Independant Isle of Man perhaps, Pictland (maybe encompassing Strathclyde and the lowlands too) Northumberland, and England and Wales as one entity, without the Anglo-Saxon/ celtic divide in OTL.