"We are professional."
Your question is a bit ambiguous - yes, plural nominative nouns only lenite their adjectives if the plural noun ends in a slender consonant, but no, that isn't the only time that adjectives are lenited.
Attributive adjectives match their nouns in case, number and gender. That means that attributive adjectives qualifying singular feminine nouns are lenited in the nominative case, and attributive adjectives qualifying singular masculine nouns are lenited in the genitive case.
On the other hand, when a plural noun ends in a slender consonant, the attributive adjective will be lenited.
There are a few other things that can lenite an adjective.
But in this exercise Táimid proifisiúnta - "we are professional", proifsiúnta is an attributive adjective, so none of this apples.
The terms "synthetic" (táimid) and "analytic" (tá muid) have nothing to do with how "natural" either form is - when taking about languages, "synthetic" mean that affixes are used, "analytic" means that separate words are used. Latin is a synthetic language, English is an analytic language.
As far as "natural" goes, táimid is more natural in the Gaeltacht in Corca Dhuibhne and Muscraí, and tá muid would be "unnatural".
'tá mo dheirfúracha proifisiúnta' is, I believe, the answer to another question in this section.
I am wracking my brain for the two terms that distinguish adjective usage... 'attributive' and 'predicative', is it? Ie, in some cases the adjective direcrtly modifies the noun ('beautiful cat'), in others it does so as a prediacate ('the cat is beautiful'). Yes?
And - irrespective of the names - in one case, the adjectives closely mirror case, number and gender of the noun, while in the other case, they do not? So: 'an bhean ghlan' but 'tá na mrá glan' ?
In English, an attributive adjective comes before the noun, and names an attribute of the noun - "the big box", "the red book", "the wet day", "the busy street". The predicative adjective occurs in the predicate of the sentence - i.e. the verb and everything after it. - "the box is big", "the book is red", "the day is wet", "the street is busy".
In Irish, both attributive and predicative adjectives occur after the noun, (for the most part - there are some adjectives that occur before their nouns). But attributive adjectives agree with their noun in case, number and gender, whereas predicative adjectives are not modified. So you have the attributive tá an bhean bheag ag siúl - "the small woman is walking", but tá an bhean beag - "the woman is small".