I will shortly begin collecting private long-term diability insurance to the tune of 60% of my former salary. Can I still apply for social security disability? If I do, would either or both of these be decreased?

Normally if you read the terms of the insurance policy they will expect you to file after a certain period.

It is up to the terms of the policy if they work together to pay equal to the 60 percent. Most will, but again it is according to the terms of your policy.
Also how long is the long term disabiity policy for?