The usual explanation is that woman is a new concept. In the past you were a caileag (Irish cailín, English maiden) until you married. Then you became a bean (Irish bean /ban/ English woman). When social attitudes changed, and an all-encompassing word was needed, Irish and English just expanded the meaning of the words, but Gaelic repurposed an old word, boireannach (originally 'something female', hence neuter, masculine in Modern Gaelic).
A very important question as people are often put off when they learn a new language by thinking 'what's the point of this?' But what's the point of learning is/are/am in English? It virtually never makes a sentence clearer by having three words to juggle with that mean the same thing, and no one (including you, I'm sure) is upset that they all translate the same way in Gaelic.
Well I don't understand the psychology, but all natural languages do seem to require this duplicate information, or 'redundancy' to use the technical term. It means that if you misunderstand - or mishear - one word, there is usually enough duplication that you understand from the rest of the sentence. So it does help with overall comprehension. In terms of whether it would actually make any sentence ambiguous if it were left out, the answer is 'rarely in Gaelic' (a 'her' and a 'his' being the main example, and even then there are ways to get round the problem), but these mutations (plural, see below) tend to be more important as you go south.
The historical explanation is that all Insular Celtic languages have them. I don't know about the extinct continental languages like the Gaulish that Asterix spoke. Yes I did say 'them'. One in Scotland but as you go south it gets worse. Two in Ireland and the Isle of Man. Four in Wales. About five (?) in Cornwall and Brittany. So it goes back to before these languages split, possibly even further - I guess that's well over 2000 years ago. That's well before writing in this part of the world (except for literally one or two words recorded in Latin texts). So everything we know is what has been figured out from later sources (with not much in Irish before the 8th century, or Welsh before the 12th century).
It's not unusual or surprising that sounds should be affected by their neighbours. This is called 'sandhi'. For example in French, the b after a vowel in Latin liber became a v in livre. We know, both from the Celtic languages, and from related words in other languages, that all lenition was originally caused when the previous word ended in a vowel, so the equivalent of my boat in Proto-Celtic became mo bhàta in Gaelic. This is the same sound change as in the French example, but across a word boundary. This is called external sandhi. Eventually this became 'grammaticalized' - that is, it became a rule in the language, rather than just something based on sounds. Once it was a rule, it no longer mattered if the word changed and no longer ended in a vowel, as the rule was stuck. So all words that cause lenition used to end in a vowel. They may no longer, but even if they don't, you can almost always find a related word in a related language that does.
Does that help at all? I am quite sure that literally millions of words have been written on this subject, so this is only a tiny start. D
Entirely a problem with Duolingo. It is not optional at all.
Note that there are some situations where the lenition is debatable, for example where the word is gender fluid, but this is not one of them. There had always been lenition here since the earliest written records of Old Irish about 1400 years ago.
I thought that was the case. I really wish that Duolingo would recognise this as a bug and correct it. I can't remember now whether this was a "Type what you hear" question, but this problem appears to me to occur particularly with that type of question, allowing some totally ridiculous answers to be accepted. I raised a bug report for it on 1st March 2020 but, apart from an email acknowledgement that the report was received, have had no response.