Yes, although the British seemed to be okay with other countries grabbing territories in close proximity to theirs.
No they didn't
They took control of 1/3 of Africa mostly to stop others taking it.
Britain mostly cared about strategic ports, but keeping these secure requires controlling inland areas too if there were rival colonial powers.
Much of the empire was a drain on resources.
The agreed partition of Africa was to avoid unnecessary conflict.
But for whatever reason, Britain had an inordinate fear of Russia beyond what would seem normal or rational, considering the more immediate physical threats closer to their homeland. I don't believe they were ever really that worried about Russian expansionism (because there's no sense to that argument), as much as they just had some kind irrational hatred of Russia.
The East India company took over India, financed by Indian elites (who saw them as more likely to pay debts than the Mughals) and using mostly Indian troops.
Even after coming under crown control, it was a long way from Britain and could only be held with the support of local elites.
A European power with land routes to India and the ability to attract local elites was a threat.
Britain conquering Afghanistan was a threat to Russian interests.
Hence the great game.
Remember this is before modern transport and communications technologies where the world was shrouded in a fog of uncertainty and it could take months or years to respond to any actions in far away lands.
The idea it was some strange Russian hatred is bizarre seeing as Britain generally sided with whoever kept the balance of power best.
Peter the Great lived in Britain and learned about ship building there prior to his expansionist career.