Ive seen people run The Sims 1 on Windows 7 before with little adjustments. I know it can work.
How do you know what someone else did to get the game running? 'I've seen people playing it on Windows 7' is not the same thing as 'I know that it was running on Windows 7 with little to no adjustments'. You wouldn't know unless you were there when someone installed it. It's more likely that you're simply assuming that you've seen someone run it 'with little adjustment'.
A lot of people who like to run older games already have virtual machines or emulators set up in their system. I've got an entire HDD that acts as a boot drive with WinXP on it and another with Windows 7 on it. I've also got a HDD that runs Linux. Plus, another HDD that acts as a shared data storage drive for my various boot drives. I don't need to emulate anymore because of this. I just switch which drive I'm booting from in my Bios when I start my PC.
I also have experience with emulators. They are a bit of a pain to set up, but once you have them in place they don't require any extra work. Once you have everything set up from the initial install it's no more complex to run a game than it is to start a modern game. Most often you just click on an exe or shortcut and start the game.
Sims 1 notorious for not liking to run on Windows 7 or Vista at all, the engine is too old and its drivers are so dated they don't run well on modern systems. It depends on systems as well, every computer is different. What might run easily on your system might run poorly or not run at all on someone else's even if you're both running the same OS.
The installer for DX7 won't run on Windows 7. However, you can try running the NT4 sp3 installer. Though, it is going to give you error messages that says you have a newer version, it may let you run the game if you install it anyway. Running it in compatibility mode will improve the chances of success if it does help.
Windows NT Service Pack Windows NT Service Packs Windows NT 3.x Service Pack Windows NT 3.1 Service Pack Windows NT 3.5 Service Pack Windows NT 3.51 Service Pack Windows NT 4.0 Service Pack Service Packs
It may be that your only option is to run the game with an emulator or not at all. I suspect that if you saw someone running Sims 1 on Win7 that was how they were doing it. I've never seen anyone run the game on Win7 without some sort of emulation going on. Though I have heard that some people managed to get it running by using compatibility mode, I've never seen it myself.