Yes. The reason why this is done is it mean that you can tell when reading the word what the word looks like without eclipsis, so 'bportán' is pronounced as if it's 'bortán' and you can tell that the original word is 'portán'. While initially confusing, it's actually a really useful feature of the orthography.
Here's more information about eclipsis: http://www.nualeargais.ie/gnag/eklipse.htm
Yes, it works similarly for c, p and t, but for b, d and g, it's more similar to the nasal mutation. Plus Welsh eclipses m and Irish doesn't have ll and rh. Some of the rules are also similar. Lenition is more similar to the aspirate mutation, even though lenition means ''softening'', so they clearly have different ideas of what softening actually is.
There are notes on it here: https://www.duolingo.com/skill/ga/Eclipsis
The gist is that when eclipsis is in effect, unvoiced sounds (f, p, t, c) become voiced (bh (v), b, d, g; which are written as 'bhf', 'bp', 'dt', 'gc' to preserve the original spelling), and voiced sounds (b, d, g) become nasalised (m, n, ng, but written mb, nd, ng to preserve the original spelling).
Now to dive deeper in to the sciencey bits.
As to why this happens, it's generally that the last letter in the previous word used to be a nasal sound, such as 'n', 'm' or 'ng'. When this final letter disappeared, it left its mark on the word that followed it.
You can see this most clearly if you look at the preposition 'i', which means 'in'. Historically, this was 'in', just like in English, but the final nasal consonant was mostly lost over time (though it was preserved when 'i' comes in front of a word starting with a vowel), so whereas once you might've said 'in portach' to say 'in a bog', now it's 'i bportach', as the nasal sound has been lost, but its echo is still seen in the eclipsis of the word that follows it.
As to why eclipsis is used with 'ag' as in 'ag an bportán', that's because technically 'pórtan' here is in the dative case (which you mostly don't have to worry about except with certain nouns as it's identical to the regular nominative case 99% of the time). The dative case is the case used with virtually all simple prepositions, including 'ag'. Where knowing this is important is with the 'an': while the nominative and dative 'an' mostly look alike in the modern language, in older versions of the language, they were different, and the dative 'an' triggers eclipsis.
To give you an example in English, once upon the time, the indefinite article was always 'an'. If English had progressed the way that Irish did, then we would have 'a' and 'an' (before vowel sounds), but you'd write 'the cat' and 'a gat', 'the boy' and 'a moy', 'the goat' and 'a ngoat'. Much the same thing happened with the dative definite article in Irish, hence 'ag an bportán'.
 A simplification, but essentially true.
 This process happened over a millenium and a half ago, back in the 5th or 6th century, back in the Ancient Irish era, so it's likely you wouldn't have said 'in portach' but something not exactly dissimilar. I don't have an etymological dictionary to check what it actually might've been based off of the Old Irish spelling, unfortunately.
ag is the basic form used with nouns, e.g, ag an gcailín, ag an mbuaichall, ag an mbean, ag an bhfear.
aici/aige is combining "ag" = "to have" with the pronoun "she" --> she has and aige is with the pronoun "he" --> he has (this is explained in detail in a later section).
In Irish most prepositions are usually written on their own, but when you use them together with a pronoun (me, you, he, she, it, us, them), the two words get contracted together to make what are known as prepositional pronouns
It might look like it, but it's not. The super literal translation of this (though keeping English word order) is 'There is water at the crab'. So the order isn't backwards (aside from the verb coming first in Irish, and that the verb 'bí', when used intransitively, is used to make existential statements), but that Irish doesn't have a direct equivalent of the English verb 'to have', and possession is instead done using the phrasal verb 'bí + ag'. Thus what is literally translated as 'there is water at the crab' actually means 'the crab has water'.
This might seem weird, but this exact construction actually occurs in a lot of the world's languages, Finnish and Russian being two European examples other than the Celtic languages. In fact, there are some interesting things to note about languages with 'have' and languages without 'have': http://www.mitpressjournals.org/doi/pdf/10.1162/LING_a_00076
why not just say, Tá an portán uisce"? Is it not the same thing? The way it is written, it almost sounds like the water has the crab! How would you say, "the crab is in the water"? It seems it would be damn close to this sentence. I was doing great with this course, until this confusing chapter!
ag, ar, le, roimh etc are called "prepositions". In English, words like "at", "on", with", "before" are prepositions. They are used to indicate relations between nouns.
A "pronoun" in English is a word that substitutes for a noun. "I", "me", "you", "he", "him", "it" etc are all pronouns (there are many classes of pronouns).
In Irish, when the object or complement of a preposition is a pronoun, they combine together to form a prepositional pronoun.
ar an mbord - "on the table" ("the table" is a noun, it doesn't combine with ar)
orm - "on me" (mé is a pronoun, so it does combine with ar to produce orm)
roimh an bportán - "before the crab" (noun, so no combining)
roimhe - "before him/it" (pronoun, so you combine to form a prepositional pronoun)
The added complication in this case is that Irish, like Russia and Hindi, doesn't have a verb for "have", and instead uses the verb tá and the preposition ag for this purpose, so tá uisce ag an bportán means "the crab has water", but tá uisce aige means "he/it has water", because ag doesn't combine with the noun an portán, but it does combine with the pronoun.
Grammatical terms are just labels - you can speak English without knowing those terms, and you can learn Irish the same way that you learned English, by trial and error, bit by bit. But if you want to leverage your knowledge of English to help you learn Irish, learning those labels can be helpful, so that you can use your understanding of how "verbs" work in English to apply to "verbs" in Irish, and you can understand why different rules might apply to an "adjective" rather than a "noun". Other people won't take the time to explain everything in very basic terms every time you ask a question, so if you want to use the help that other people offer, make the effort to understand the terms that they are going to use. People aren't going to explain the difference between a "vowel" and a "consonant", they'll expect you to look it up yourself.