Does anybody know whether it is mandatory to have a guide for the Milam Glacier trek in Uttarakhand, India? What about permits?
Milam glacier falls under the Nanda Devi Biosphere Reserve limits. In case you want to enter a national park limits, you will need a permit from the forest department. Additionally, since parts of NDBR fall under the protected areas act, you'll need a protected areas permit to enter the Milam glacier section. The PAP can be obtained from the MHA, Dehradun. The other option is to contact the forest department of Uttrakhand and see if you can get the permit from some other source. Ideally only a PAP should cover both the protected areas permit and the forest department permit. If not, you'll have to additionally get the forest permit from Joshimath or Dehradun.
Most of the national parks in India mandate a guide as this would prevent the tourists/trekkers from wandering off the set trails. I do not see why this would be any different in NDBR (Nanda Devi Biosphere Reserve) as this is a UNESCO world heritage site. Most of the guides for the NDBR are located in Joshimath.
The other option is to directly contact a trekking agency in Joshimath and allow them to arrange for both the guides and the permit for you.
Also, please note that the national park is open only between May and October.
I did this trek in 2012 on a guided tour organized by IndiaHikes. There was a large flood in 2013 in Uttarakhand which, from what I've heard, affected the trek significantly and made it harder.
As to your question, a few pointers:
The trek itself was relatively easy back in 2012. If I was more familiar with trekking in India, I might have done it without any guides.
The basecamp for the trek was at Munisiyari. We took the Shatabdi train from New Delhi to Kathgodam railway station. Leaves at 6:00 and arrives at 11:40. From Kathgodam, we stayed a day at Nainital (cabs are available right outside the station. 1.5 hrs approx. to Nainital).
We took a cab from Nainital to Munisyari which was nearly a 8 hour, gruelling ride (~265km). There are also buses from Nainital to Munisyari which are cheaper though uncomfortable.
We stayed at Pandey Lodge which is quite popular among both local and foreign tourists. You don't have a lot of other options, really.
The actual trek is about 14 days if you want to go all the way to the glacier head.
You need permission to reach the glacier. Our tour guide had already arranged them so it wasn't a problem. However, I've read a report that tourists are currently being denied permissions because of damaged tracks, though the accuracy of the news source seems dubious. You'll probably have a better idea if you call the hotels in Munisiyari.
As to whether you can do it on your own, I would ask: how familiar are you with India in general and Uttarakhand in particular? You need permission to enter the glacier head in the Nanda Devi Biosphere Reserve. From my experience of dealing with Indian bureaucracy, it can be hard to navigate unless you're familiar with India.