1)Yes, an international standardized character alphabet exists for transcribing the sounds of all human Languages. It's called the International Phonetic Alphabet and it is maintained by the International Phonetic Association (both are acronymized as IPA). The most recent version of the alphabet was created 1969 and their most recent and currently operative handbook, first published in 1999, is the Handbook of the International Phonetic Association: A Guide to the Use of the International Phonetic Alphabet. Although as a guide, it's rather useless as I explain below, and certainly is not for beginners. If you need to learn the IPA, pick up a good introductory linguistics textbook, it should teach you the IPA and any background info you need to understand the alphabet, like articulatory nomenclature. If you aren't already familiar with baisc linguistics terminology, then the Handbook will be a very difficult read.
2) There is a concept called orthographic depth which is a measure of how well a language's orthographic symbols are in 1-1 association with the language's phonemes. Phonemes are the finite discrete combinatorial units of sound that are unique to a given language, though all languages draw their phonemic inventory from essentially the same superset of phonemes since all humans have the same speech organs (tongue, larynx, alveolar ridge, etc.) and it is our speech organs which define what sounds are even possible. The entity being represented and transcribed by an IPA symbol is a phoneme. It is somewhat a curious thing how every language, including Japanese, possesses a highly intricate and constantly changing set of rules that govern how these phonemes can combine to produce the verbal form of words. This set of rules describing how the phonemes pattern to form the words of a given language is called a phonology for that language. Languages with the same inventory of phonemes can have different phonologies ie different rules in how those phonemes are combined to produce words. IPA symbols representing phonemes are concatenated and placed in between two '/' characters to represent the phonemic representation of words. So, <今日は> is phonemically transcribed as /koNːit͡ɕiɰa/ using the IPA. A sequence of phonemic characters is called a string, a term borrowed from computer science.
However, even though there is a standardized alphabet, the transcription for a given phoneme, sound, or word can differ depending on how you use the alphabet. The IPA is only an alphabet, it doesn't tell you how to interpret and transcribe what you hear. It is still left to devise a scheme for associating the IPA symbol to the phoneme you are hearing. And there is usually several mutually contradictory schemes that you could define for transcribing the phonemes of a given language's phonemic inventory that are all consist with the IPA. It usually comes down to how you segment the speech stream. When you study phonetics, you'll know what I mean. If you look in the Handbook at the languages they exemplified, you'll notice it's referred to as an "illustration of the IPA for language X". This is because there are different ways to transcribe with the IPA. For example, sometimes you can add diacritic marks to make the transcription more accurate, but only if you really need that extra detail.
The IPA is only for transcribing phonemes, it says nothing about how a language is to combine those phonemes to form words. In other words the IPA is agnostic to any particular phonological theory.
Most kana are in 1-1 correspondence with pairs of phonemes, but several kana are 1-1 with singular phonemes. So, <む> corresponds to the phoneme pair /mɯ/ but <あ> corresponds to the phoneme /a/. <ん>, however, is a little more complicated to describe. It correlates with an underspecified phoneme, a special type of phoneme that the IPA doesn't accommodate because it's a language dependent phonological conception. There's no room to explain that here. Because most kana are in 1-1 correspondence with a single or pair of phonemes, kana is said to have a shallow orthographic depth, but it isn't perfect 100% 1-1 correspondence when considering the entire Japanese script and it also depends on how you define orthographic depth. These technical definitions are important if you are trying to do something like getting a computer program to read Japanese text and synthesize speech. Such a task is called grapheme to phoneme (G2P) conversion.
Kanji on the other hand is more or less arbitrarily associated with phonemic strings. In other words, when you encounter a novel kanji for the first time, you cannot predict its phonemic correlate ie how it is pronounced. Because of this Kanji is said to have a deep orthography.
You can type out IPA symbols then copy and paste them however you like using this website http://westonruter.github.com/ipa-chart/keyboard/ . This is how I type IPA to include in web pages. You need to set your browser to view UTF character encoding to see IPA characters. Do that if the IPA characters are not rendering.
Since kana is orthographically shallow you can construct a list that maps kana to IPA phoneme strings rather easily. But you need to establish your transcription conventions beforehand, and to what level of detail you need to transcribe. For these reasons, a phoneme to kana implementation is usually created only for the application at hand.