Sound bars are designed to connect into TVs, so it should have some sort of other input connector on it. As why_wolf has stated, have you checked to see if the bar has an optical connector?
Something like this diagram below
It sounds like your will need to run a system repair. Use Windows 10 Install media to go into windows setup and try to run a system repair.
See here for more information
You would want a minimum of an i5 Processor, but really your budget is restricting you from getting anything that will perform well in games. You would have to get a laptop with a good GPU other than an Intel HD iGPU. So really this budget would be moving into the $750-900 range. (3000-3500lei)...
Windows 7 updating process seems to be completely borked now. If I am re-imaging any systems with Windows 7, I have had to ensure that I have the latest updates or the "convenience rollup" slipstreamed into the disk image, so that I don't encounter the laborious update process for an SP1 ver...
Go into your bios settings (pressing F10 I think on HP notebooks) and see if the hard drive is being detected. If you can see it in bios, but it still wont boot, then there's a good chance that it is faulty/failed and will require being replaced.
Nowhere have I said that TV's have HDMI output. I was referring to the projector possibly not having an HDMI input, and making OP aware of the factor.
Nor did I say that the display affects the ability to run the OS.
You're too quick to slate somebody else's advice.....it's not a competition...
Best bet is to plug the Firestick into a LCD TV or any device with HDMI input, then connect the LCD TV to your projector. I don't think your projector will run the android based OS directly from the stick. There needs to be HDMI input. (not HDMI output)
So when you first switch the laptop on......are you getting any display before it starts to load Windows? It should at least be showing you a post screen to allow you to access bios.
Did you try plugging in an external display yet?