Is it obligatory to make up the prayer immediately upon returning home, or after eating when hungry and how that affects khushu' (devotional humility), and which is better: performing the prayer individually on time, or in congregation with delay?

If your delay in performing the prayer leads to its time expiring entirely, then this is a grave evil and a major sin. You must repent from this sin and be diligent in performing the prayer on time. However, if the delay is only from performing it at the very beginning of its time, there is no sin in that, because performing the prayer at the beginning of its time is merely recommended. Prioritizing food over prayer is permissible if you are in need of it and your hearts would be disturbed by leaving it, provided that this does not lead to missing the prayer's time. As for which is better: praying individually at the beginning of the prayer's time or praying in congregation after the time of preference has passed, praying in congregation is better, even if it is after the initial preferred time, unless waiting for your companion leads to the prayer's time expiring entirely. In that case, it is not permissible, and you must then perform the prayer on time, even if alone.