Akbar's Dīn-i Ilāhī was a pretty laudable attempt to combine the best parts of many religions. Unfortunately it did not take off very well. To answer your questions:

Did people willingly convert?

His inner circle - that is those of his court who were very close to him did convert

Was it a genuine religion or merely an idea on the part of Akbar and his "cabinet"?

This is a lot tougher to answer as then you would have to go into the question "What is a genuine religion?" But for the purposes of this discussion I think it would be correct to conclude that while Akbar wanted it to be a popular religion, it never really reached the masses.

Did it survive past his reign?

Unfortunately, no. The best of Akbar's religious ideas died with him, though his general administrative philosophy of including the Hindu Rajputs continued for his successors.

Was it influential to India's subsequent religious or political history?

Well, not Dīn-i Ilāhī itself but it can be argued that the broader concept of finding things of value in other religions and for the Emperor to look beyond religion in his administration did persist for quite some time.

Edited for sources:

J.F. Richards' book on the Mughals is a pretty good source, though you should really be looking at Indian authors too. Irfan Habib has for pretty long been a pre-eminent authority on Mughal India, Athar Ali is also someone to read. A very scholarly treatment of the Mughal state is Sanjay Subrahmaniam's The Mughal State