FPGA System Integration
How companies integrate an FPGA into a system can vary a lot from one system to the next due to the flexible nature of an FPGA. No 'special' FPGA is needed because the FPGA is inherently programmable. A company is free to program into it whatever interface best suits their needs, whether that be a complex bus like PCI or a simple interface like SPI or I2C. Some additional ICs may be needed external to the FPGA to support the desired interface, but this a meek point.
The question is sort of like asking how software companies integrate they're software in an operating system. Does it interfaces with a database? If so maybe it will use SQL queries. Does it have online capabilities? If so, maybe it will access web services. In any case, it's a very open-ended scenario.
The latest trend, however, is to integrate processors and FPGAs on the same SoC. In this case an interface named AXI (Advanced eXtensible Interface) is used between the processor and programmable logic. AXI uses a topology similar to PCIe but specialized for high-performance on-chip communication.
High-Level Programming
This is possible. It's called high-level synthesis (HLS) and allows the programming of FPGAs from C code. HLS, however, has it's limitations. For one thing, it's been around for ten years now and only very recently starting to gain more traction. But ultimately, systems with FPGAs will tend to use a combination of high-level languages (C/C++) and RTL (Verilog/VHDL). High-level languages are used to program the processor while RTL is used to program the FPGA. And now, optionally, C can also be used to program the FPGA through HLS.