I think that it's Important to emphasize that there are many different forms of communism, and that not all countries were able (or even found it desirable) to use the same model.
Marxism, on which communism is considered to be based, operates on a basic fundamental principal that the society transitioning to communism has a industrial base in which the producing classes were being exploited by the non-producing classes.
China was primarily an agricultural nation in which the producing classes were farmers rather than industrial laborers, and in where the exploitation was by a non-producing landlord class, not a non-producing management class.
Therefore Marxist based communism could not be achieved because the foundations were not in place, and were not desirable to be put in place because they aimed at resolving problems that China did not yet have while not addressing the issues that it did have.
To address each of your issues directly:
- Abolition of private property, everything is public.
This was the original aim, but China saw that this policy was failing in Russia and was producing inefficiencies, while it also failed to address the aspirations of China's landless rural population (Which provided the mainstay of Mao's early supporters).
China moved farmland and industrial property into public ownership, but maintained a quasi-private system in which land was allocated to people rather than collectives, as this provided a greater level of motivation and cultivated a sense of ownership.
In recent years China has moved away from this, and has allowed increasing private ownership, including corporate ownership of farm land.
- Progressive Tax.
I am not qualified to address this issue.
- Abolition of rights inheritance
China allowed the inheritance of property and wealth, and the transfer of the right to land, though the land remained in public ownership if it was agricultural and only permission to farm was transferred. This provided people with greater motivation to be productive
- Confiscation of property of all emigrants and rebels.
This is too long to answer here
- Centralization of all credit in the hands of the state.
China has state owned banks, private banks have recently been introduced based on the reforms of Deng in the late 1980s due to the state owned systems becoming stagnant and inefficient. Though the state maintains a controlling share in many institutions that is great enough for governments such as Washington to consider them state owned.
- Communication and Transportation in the hands of the state.
See 5
- Factories and instruments of production owned by the state.
See 5
- Equal obligation of all to work
To the extent that is possible, yes.
- Abolition between distinction of town and country by equable distribution of population over land.
This was attempted but failed, and was later moved away from once the populations of cities became substantially more productive.
- Abolition of child labor; child rights to education; education combined with industrial production.
This was, and still is, being attempted.
To summarize, China attempted to follow the principals of Marxism but with the emphasis on the rights of landless rural laborers rather than uneducated industrial laborers.
It attempted to redistribute land, property and wealth to the above population, but found that rural laborers made poor industrial laborers, and vice-versa. Causing a drop in agricultural and industrial production that lead to recession and famine.
These policies were abandoned or adapted under Deng during the 1980s, and the position of the state was changed form being an owner to a stakeholder, but not to the extent that China is considered to be a free market economy by the US.
While this cannot be considered to be pure Communism based on the principles of Marx, it can be considered to be Communism with Chinese characteristics because it fulfils (or at least is intended to fulfil) many of the aims of Marxism, but not necessarily in the ways envisioned by Marx. To quote Deng "It doesn’t matter if a cat is black or white, so long as it catches mice".
