He had promised all power to the workers Soviets in his April theses and temporarily went along with the idea of a constituent assembly, but communism was slower than Lenin had wanted:
Did the single-party state simply arise from the fact that socialism wasn't yet fully implemented yet in Russia, and the Communist party would need to stay in rule to embed it?
Edit: Moreover Russia had been in a state of political flux for so long that economic issues had not been dealt with. Perhaps the single-party state was considered necessary until economic issues could be resolved.
Were there any factions within the party (or any other single party system's party) that functioned as de facto parties?
No, the Vanguard Party model (as it is known) was specifically put forward by Lenin. Speaking as a Leninist myself, the idea of the Vanguard Party is that it begins as a subset of the "class-conscious" (i.e. revolutionary) section of the working class which simultaneously educates their fellow workers as well as agitating for revolution within the structure of the capitalist state. When revolution arrives, the Vanguard Party leads the workers in combatting the capitalists, and when the socialist state is established, the Vanguard Party organizes the state in such a way as to be as interconnected between the government, the workplace, and the general public as possible - the idea being to create a government that was subservient to and a part of the working class, from top to bottom.
To give a rough outline of what such a system looked like, in the workplace (such as a factory), workers elected a representative to local workers' councils (Soviets, where the Soviet Union gets its name). These in turn elect regional legislatures, which send representatives to the legislature of their SSR, which in turn send representatives to the Supreme Soviet, the national legislature. In addition, all workplaces, as well as schools, hospitals, and other infrastructure-esque offices, also had Party representatives, to which any citizen could go to submit suggestions or complaints. Via the process of Democratic Centralism, which Lenin described as "freedom of discussion, unity of action" if I recall correctly, these issues could be brought before the Party, who would discuss and debate the issue freely as they so wished, and then cast a vote on the matter. Party members were expected to uphold the decision, no matter what it was. Lenin famously said that this model was "1,000 times as democratic as bourgeois (capitalist) democracy".