Theme: Through faith in Christ we are brothers and heirs; live in the freedom that Jesus has secured

What do elementary things refer to?
Most likely, 'elementary things' refers to the rules and regulations by means of which both Jews and Gentiles attempted to achieve salvation/ensure future life with God in glory. In Galatians 4:9, Paul is primarily referring to Gentile believers who have come out of paganism.

How did a bodily ailment result in Paul preaching the gospel to the Galatians (cf. 4:13)?
The book of Acts doesn't testify to any sickness on the part of Paul during his first missionary journey (Act 13-14). It is not a complete account of all the places he visited, but selective, containing only what Luke considered to be essential. From Paul's words in this passage, it appears as if the decision to preach in Galatia was prompted by an illness of some kind. The most common theory is that he contracted Malaria whilst in the lowlands of Perga (Pamhylia), and that he was encouraged to go up to a more mountainous area in response (Galatia).

Whatever the sickness, it was not a liability for the spread of the gospel. Instead, the weakness of Paul became a means by which the strength of Christ was made manifest (cf. 2 Corinthians 12:7-10). There will be many occasions over the course of his life where his sufferings were used by God to spread his truth (e.g 2 Corinthians 1:3-11; 2:14-15; 4:7-12; 11:23-29; 13:4; Colossians 1:24-29).