the marriage solemnised as per the the religious rituals and caste customs are considered to be legally valid marriage i India and it is immaterial that whether it was registered or not. However the marriage certificate is essential for travelling abroad in the status of married couple
1) Would it be possible (and is it legal) for us to also register our marriage in India with June 2014 — the date of our religious marriage — as the date listed on the certificate?
With the proof and evidences of solemnisation of marriage in India, it will be possible to register marriage in India even now and the date of marriage will be the same one on which it was solemnised.
2) If 1) not not possible, can I use my US marriage certificate with June 2015 date to get a marriage certificate from India with the same date?
Not necessary, you can refer the above answer.
3) Is it illegal for us to have two marriage certificates from two different countries with different dates?
Actually a marriage certificate issued by a foreign country for a marriage solemnized in India as per Indian laws is not valid in India. So possessing any amount certificates in this regard will be of no use, dont try that gimmick.
The authentic certificate for proof of marriage obtained from US will be valid in US, in India, you dont need that certificate since your marriage has proof with the photographs and invitation card etc.