How can i find out why the feature likght on my nec dtu-8d-1a phone is flashing?

SOURCE: How Can I check voice mail remotely

Do you mean check your v/m for your extension from somewhere other than your office by calling in?
If so, it would depend on how calls are answered at your office. If someone answers, have them transfer you to the ext number for VM. If you have an auto-attendant, or a DID directed to your extension, call and wait 'til VM answers.
Assuming the default settings are in place, dial "9" + your ext number, then your security code if you have one.

SOURCE: caller ID Display

This one could have different variables.

1. your CO lines must have CID
2. your CO card in the system must have CID capability
3. older systems have a limited number of extensions that can display CID (16)
4. your ext must be set to ring.

SOURCE: why is the mic "lit"

when the light is lit it allows hansdfree use of your phone.

SOURCE: Programming speed dial numbers into NEC DTU-32D-1A phone

hit feature key,hit a line key,dial 9 plus the number,hit feature

SOURCE: NEC Directory Button on DTU phones

You should have soft keys in the display. If you look in the display and see words at the bottom of it, than the keys under each word will do the function it says. One should say DIR, this is the directory key and should show caller id, or one should say cid. You can prgram a scroll key but need to know how to enter program mode and program the feature. ldtsnj.com