You didn’t address any of my concerns, nor was I talking about productivity. Let’s try again for the the first one with a simple example:

Company 1 makes a product (let’s say timber) at 50 surplus value. That 50 is a cost for company 2 that uses the product as an input material (it makes wooden chairs). We can calculate the total rate of profit of both companies. Now company 1 is forced to lower the price to 40 because of competition. We calculate the total rate of profit again and the total rate of profit has actually increased.

Thus, it does not follow that lowering prices/profits leads to a decrease in the overall rate of profit

Competition forces prices down, and rates of profit with it

This is not true in the general case. If prices for input materials are down, profits rise for the company using them. One company’s profit loss is another’s gain. That is even with the shaky assumption that competition can exist long term in a free market. Imperialism, as defined by Lenin, results in concentration of capital and the removal of competition.

this process can be struggled against by expanding markets or finding new industries

There are counteracting forces for it, but expanding is not one of them. Expanding does not change the rate of profit (profit/capital invested); at most, it changes the total profit.

There is no tendency of the rate of profit to fall. The theory is inconclusive, as is empirical research. If TRPF were true, then growing a company would, in fact, accelerate the process.