Bismillah - meaning
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The essence (لب) of the Qur'an is Bismillah-ir-RaHmaan-ir-RaHeem بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ. It's on its start and all the rest of the Qur'an is an exposition of it. What does it mean?
A proposition bi- has the same functions as English by. It always point to a reason (cause) and/or a tool (way) for something to happen/be.
Word ism is different then in all other occurancies, for example in اقْرَأْ بِاسْمِ رَبِّكَ الَّذِي خَلَقَ, missing elif at its start. This is the case in 1141 also in بِسْمِ اللَّهِ, but er-RaHmaan-er-RaHeem doesn't stand here. We can see that ism with Allah is different with a proposition bi-. Word itself comes from the root which describes raising something to be seen clear, good, but the point is not in raising, but to clear vision.
By seeing something our mind creates a picture of it. That picture is a name of a thing for us. Translated to a language it becomes a noun. So noun is a verbal name. Name and noun are both ism in a language of Arabs. The Qur'an is a tool for us to reverse the process to create names from nouns, in order to see. Since it's Allah's ism then we have to create names from nouns about Him, in order to see Him. This is the essence of the Qur'an - to see Allah, by His esmaa' (plural of ism). That's why I think ism here stands for a process of naming, not just name. Naming here doesn't mean to give a name we'd like, as in a verb semmaa سَمَّىٰ but to understand the name He gave to Himself.
Then we see His first name, by which we can see Him. It has one verbal noun with intensified meaning and an adjective, both prefixed with el-. These are from the same root r-H-m. This root describes a process (Harf mim) in which some interrupted movement/life (harf Ha) continues or repeats (harf ra). In relation to us (readers, hearers), it's the help provided in situations we cannot find a solution, due to which we're stopping. His raHmet is help provided to us, besides our natural abilities. His speech in the Qur'an is a new creation, as His help to us who can understand it in seeing Him though His names.
Er-RaHmaan is describing Him as the Provider of permanent help (in a situation we cannot solve), and er-RaHeem is describing Him as the Provider of help in everything (in a situation we cannot solve). As the name, this describes Him as the Provider of every permanent help.
So, Bismillah-ir-RaHmaan-ir-RaHeem is telling us that what follows is for naming Him by His names, and He's the Provider of every permanent help we need in that. That's why mentioned example in 961 talks about learning by our Rabb's name, not learning random things. Rabb is a provider, and in this case He's the Provider of our knowledge, ie. Teacher.