So, I'm going to describe this in the way my class was taught. It'll be watered down, because y'know, we were first semester Irish students without much grammar knowledge. So, your word order is going to be VANTP.
V = verb
A = Indefinite nouns
N = Names
T = Definite nouns
P = Pronouns
So that's your general order. If you use a definite noun, a proper noun, or a name, you need to include the pronoun (é, í, iad, srl - note, after ní, these prefix an "h") before the first one.
So in your first sentence you have and indefinite noun (araicnid) and a definite noun (an damhán alla). So, putting these in order (A before T), you get Is araicnid an damhán alla. However, you need the pronoun before the definite noun. Since damhán alla is feminine, you use í. This giving you Is araicnid í an damhán alla.
Your second one you have a name (Pól) and a definite noun (uachtarán na hÉireann). So, putting those together, you get Is Pól uachtarán na hÉireann. Yet, you still need that pronoun. Is é Pól uachtarán na hÉireann.
Hope that helps explain it a little. And note, this is really basic, and barely scratches the surface of the copula.
In general, you need to memorize gender and declension. teanglann.ie is a great dictionary resource that will tell you this.
Despite that, sometimes you can guess the gender of nouns to a pretty high degree of accuracy. One heuristic I use is that nouns that end in a slender consonant (that is, preceded by a slender vowel like i or e) are usually feminine.
You can find a more detailed guide to gender heuristics here: https://thegeekygaeilgeoir.wordpress.com/2017/08/28/making-sense-of-irish-gender/
There is slightly more concise description at http://nualeargais.ie/foghlaim/nouns.php
The article that you referred to does suggest that single syllable words ending in -cht are masculine and multi-syllable words ending in -cht are feminine (which is more detail than the page that I linked to), but then it says that bunreacht is an exception, because it's a multi-syllable word that is masculine, but bunreacht is a compound word, taking it's gender from the single syllable reacht .
Thanks. Could you elaborate a little - in what way does the meaning change? Is that only a matter of emphasis?
I tried to find some info online and turns out that Wikipedia adds in another detail that I missed (https://en.wikipedia.org/wiki/Irish_syntax): 'If the predicate is indefinite, it follows the copula directly, with the disjunctive pronoun and subject coming at the end: Is múinteoir í Cáit. "Cáit is a teacher."'
I guess in this case it's not really indefinite as there is only one president of Ireland (shouldn't there be a definite article?).
There is an article - the na in Uachtarán na hÉireann. Since it'seems a genitive phrase, the article goes between the two.
And the meaning changes from Paul being the subject to the President of Ireland being the subject. So instead of labeling Paul as PoI, you're labeling the PoI as Paul.
I guess it must be a phone thing. I tend to go back and forth between the website and the phone app. The exercises give you five Irish words to match to five English words. They also lowercase "pól" in the exercises where they give you an Irish sentence and you translate it by selecting the English words from the boxes below the sentence.
I've no idea what you think [š] sounds like (it's not part of the standard IPA character set, as far as I know), but the wiktionary entries that include pronunciation guidance suggest that in many languages it's what we call a slender s in Irish. In which case you would be wrong - the s in is is generally pronounced [s], with a broad s, despite the fact that it is next to a slender vowel.