Both forms exist, and we use them to refer to a situation which you are waiting to happen. For example: 'He's sent off an application for the job, but he's yet to hear anything.' This means he's expecting to hear something, but he hasn't heard anything yet. or 'They've put a lot of effort into the advertising campaign, but they've yet to see any results.' This means that that they haven't seen any results yet. This form is simply an alternative way of phrasing a present perfect sentence ( haven't seen/ hasn't heard) by looking into the future rather than the past. In some ways, the 'am to' part of this construction is similar to the rather formal future form 'be + to + infinitive' which we use to talk about a scheduled event. For example 'The President is to visit Paris next month'. The 'looking forward' aspect of the construction suggests the idea of expectation. It's a nice expression because it encapsulates past, present and future time. It hasn't happened yet, but you are expecting it to happen at some point. That said, it is by no means an essential construction to use. Use it if you wish, but there's really no need to! Plenty of native speakers get through their lives without ever using this construction, I'm sure.

"I'm yet to..." is not a proper usage. "I've yet to..." is the proper usage. "Yet" is a word that indicates an unspecified time. We usually use it to indicate that something should have happened, but it hasn't happened (yet). When using it like this, we are speaking of something that should have happened already in the past (and maybe it has, but the time is unknown). This necessitates the use of the present perfect (here is a good overview of the present perfect: http://www.englishpage.com/verbpage/presentperfect.html), which uses the auxiliary verbs have/has. "I'm yet to..." is in the present tense. "I am yet to eat lunch." This doesn't make sense. "Am" is a present tense being verb. I cannot be eating lunch, but not eating because it hasn't happened yet! Use "has" or "have" when needed in reference to "yet." I tried to find something online that might go through this more thoroughly, but couldn't find anything off hand. Now, I don't think I could live up to your request to explain this "once and for all" for you, but I hope I helped a little!

It means 'to be about to'. "That is the hardest exam I've yet to do". (That is the hardest exam I'm about to do or that I'm going to do)

'I'm yet to' and 'I've yet to --- you used those when you want to express that you have not done or accomplish something yet