Hi John,
The integrated graphics of your i5-6400t CPU will work fine up to 1080p resolution. The i5-6400t CPU with integrated Intel 530HD graphics has plenty of power for 1080p, but for 4K, not so much. Intel HD 530 strains to play 4K resolution at it's maximum 30Hz refresh rate. It won't work at 8K. So it depends on what you want,and what your TV / Projector can do. So please advise that, with make/model numbers if possible. (Details on all potential connected components please.)
I'm with TLS on replacing the power supply, but no so much on the remaining components unless they are aged or flaky. As you do not mention pushing the audio to a sound system (other than the "asus stxII soundcard to reproduce the audio"). Your asus stxIIwith it's great specs is limited to 7.1 sound, so that may be a concern if you are interested in the latest immersive sound codecs (Think Atmos). Most folks these days use HDMI to connect to a AV Receiver, and then output from it to the Screen / TV for video. I have a PC connected to my Man Cave HT setup in this manner. Of course you can connect via HDMI from the PC to the Screen / TV.
Asus Motherboards (MBs) have been my "go to" purchase for years, as building PCs for family and friends has been a major hobby along with being into Audio. Early on their manuals were head and shoulders above the field, but in recent times they've fallen back into the crowd. Another good OEM is Gigabyte where enthusiasts park their $ / £ .
I'd recommend you buy a MB with M.2 Hard Drive capability if the Chipset / MB supports it. Buying a Gamer MB adds extra cost and features you probably won't use. But as your CPU was released 6 years ago, a matching Gamer MB of that generation, if available may be what you can readily find for purchase.
Almost all compatible MBs will come with a slot for adding a graphics card. As they are very expensive these days due to the Global "Chip" shortage, I'd build your PC and only add a graphics card if I was unsatisfied with the video results, or wanted to go into Gaming or showing native 4K or 8K sources.
At the moment I'd recommend:
ASUS PRIME Z270M-PLUS LGA 1151 Intel Z270 HDMI SATA 6Gb/s USB 3.1 Micro ATX Motherboard At Newegg
It fits your budget and known needs /wants.
I hope this is helpful.