This is wrong on both counts.

1. The large firms and key industries are overwhelmingly publicly owned, public ownership is the principle aspect of the economy. Capitalism isn't just "markets," it's when private ownership is the principle aspect of the economy. In the PRC, public ownership is principle.

2. The working class has recieved dramatic increases in real wages year over year consistently for decades. Development and planning of the economy puts the working class first, which is why their conditions habw improved dramatically. The CPC is not a bourgeois party.

For further reading:

1. Qiao Collective's Introductory Socialism with Chinese Characteristics Study Guide

2. Socialism with Chinese Characteristics ProleWiki page

3. Socialist Market Economy ProleWiki Page

4. People's Republic of China ProleWiki Page

5. My "Read Theory, Darn It!" Introductory Marxist-Leninist Reading Guide

6. Has China Turned to Capitalism? Reflections on the Transition from Capitalism to Socialism by Domenico Losurdo

7. China Has Billionaires by Roderic Day

8. The Long Game and its Contradictions

Yes we do. The PRC is in the developing stage of socialism. The large firms and key industries are publicly owned, the medium firms are heavily controlled and planned, and as they grow they are further integrated into the planned nature of the economy. This is the general path to sublimating property and gradually collectivizing it. The US Empire, on the other hand, has private ownership of the large firms and key industries, and runs on imperialism, ie vast expropriation of wealth from the global south via financial and millitary domination.

I suggest reading Marxist-Leninist theory, if you want somewhere to start, I wrote an introductory reading list targetting those interested in theory but who don't know where to start.

The overwhelming majority of practicing Marxist-Leninist orgs support the PRC, and Marxism-Leninism is in turn by far the most significant branch of Marxism. Marxism is more broadly adopted than anarchism globally, the other major school of thought in the socialist movement.

The PRC does support the PRC, yes.