Can anybody explain why ‘szczęśliwego’ (masculine adjective, gen/acc case) is used in this sentence rather than ‘szczęśliwy’ (masculine nom)? Is it because of the verb governance of ‘wygląda(ć)’?
There are several sentences on CM where wyglądać (in its conjugated forms) is teamed with szczęśliwego.
Or has the adjective been changed by declination because of ‘na’? Thanks in advance.
It’s the governance of the preposition “na”—in this context, it requires the accusative form, which is “szczęśliwego”.
In general, prepositions take precedence before most (if not all) other parts of speech when deciding what case is required.
When you have a combination of a verb and a preposition, it is always the preposition that governs the case of what follows. Keeping this rule of thumb in mind will make your life much easier 
Here are two examples of such combinations:
If you need a refresher on prepositions and the way they force other parts of speech to take the appropriate case, you can take a look at this blog post: https://www.clozemaster.com/blog/polish-prepositions/
Thanks! I think I must have had a ‘brain freeze’ when I asked this question! Of course, a preposition will take precedence in determining the case of the following word. It’s a pretty basic rule that I temporarily forgot! I’ll definitely have a refresher by looking at your blog. Cheers