Did the British-German-Russian alliance happen before or after France began heavily investing in the Russian economy?
Before, Russia is considerably weaker in this scenario. Since the Germans have only given token investment and he British are well the British, they don't like the Russians but
Is the BGR a formal or soft alliance? Is Britain committed to sending troops to the continent, or are they simply favourably disposed to the German-Russian alliance?
Soft, Germany is in a formal alliance with Russia and has a separate mutual defence agreement with Britain, specifically aimed at France where Britain joins the a war against the French is they declare war on Germany first, but stays neutral in any other situation, it also has a guaranteed protection for none military shipping through the channel if a they are in a war with France and Britain is neutral (which is an easy casus belli for Britain to enter the war if they want).
Does the Entente Cordiale still happen, or do France and Britain have unresolved colonial issues?
No, both France and Britain resolved their colonial dispute similarly to the Cordiale, with France giving a minor concession in Vanuatu which became a British protectorate rather than a condominium with France.
What of Austria-Hungary in this scenario?
Minding their own business, they've become neutral and are dealing with some unrest. Germany basically threw them under a bus and supported the Russians in the Balkans, but in their alliance it was made pretty clear that they will both support a unified AH against any internal rebellion.