How does one become a courtier, with lodgings and all? Also, in the cases of Lords Paramount living at court (Renly, Stannis) who rules their respective realms in their stead?

Well, you need to be highborn, to start with. 

Beyond that, it’s a matter of patronage – the office you’re occupying to get the lodging and the food and the livery is a favor that the person in charge can dole out or not as they desire. So it’s about who you know, and who knows you – Ser Andrew Estermont, for example, was Stannis’ squire at one time because he’s Stannis’ kin. Historically, bribery was also involved – send someone a nice horse or a hawk or some nice tapestries, hey presto your supernumerary relative can have a job! 

Beyond that, you also need money. While your office usually comes with room and board and a small income if you’re lucky, you need to be able to mix with the upper crust and live up to their standards. That means fancy clothes, the right kind of horse, hawk, and hound, money for gambling and booze, servants to cook and clean and do all the work for you, etc. 

As for Lords Paramount, it’s usually either a relation (as in the case of Robb and Cat taking over for Eddard) or the castellan of the seat of the region (Ser Courtnay Penrose).