The ASC appointment date we got is 3 weeks from now; can we do an early Walk In? [W]ill they do it for UK visa applicants?
It would depend on the ASC, but most allow it, although you should call to be certain. If you can, try to arrive at opening time, but still be prepared to wait; some centers are small, most are busy.
Do we need to courier the applications separately to UKVI, or we can send it together, as we will be traveling together?
While you will be travelling together, both applications will be witnessed, logged, and evaluated separately. If you applied through Visa4UK and are sending them to UKVI Consulate General in New York, you could send them in one package, each in its own envelope. The problem it presents it is managing for the return of your passports and supporting documents, as the return label would be affixed to one.
If you've gone through VFSGlobal, its courier service does allow you to send them in one package (as passports and documents are returned to VFS, and it has a tracking/pairing system in place).
Also refer to UK Standard Visitor Visa: How to mail multiple applications