DSP processor: some beginner's questions
I'v have some experience with TI C55 and C6000 series DSPs, I'll give you some points, hope to be helpful.


  Programming each chip that came from a different company requires different softwares or big differences in commands?


Absolutely! Although the code body might be the same, but since every company has its special processor architecture, their compilers would be different. Even two DSPs of a same company might be too different, for example Texas Instruments C5000 processors are too different from C6000, since their memory architecture is different. I mean, you can't simply use the same segment of code for both.


  Is there a company making both DSPs and Bluetooth?


Usually most of the DSPs are only Microprocessor and a few essential Interfacing (like I2C, SPI,...) are considered for them, however some are more sophisticated System on a Chip designs might include Bluetooth. Basically you can buy and add a Bluetooth module to any DSP you want.
https://www.sparkfun.com/products/12574


  If you have some tips regarding the chips I will be happy to read them :) (easiest way to begin, where I should go into the architecture and what is less importent and can look at it from a bigger picture)


It depends on your programming level and those calculations you've mentioned above, however, I think the best chip to begin with is MicroChip DSPIC, which is simply a PIC micro associated with some DSP blocks (like circular conv, FFT ,etc), I think your last choice should be TI's DSPs or AD's Shark DSPs, they are too complicated and frustrating to begin with.