Is it permissible to combine prayers individually after the funeral prayer if one caught the second rak'ah of the Asr prayer, which was combined with the Dhuhr prayer by way of taqdeem due to rain?

If your friend joined the congregation with the intention of praying 'Asr, he has disrupted the order, and this order is waived due to ignorance according to some scholars. If he joined them with the intention of praying Dhuhr, some scholars have permitted this. The majority of scholars stipulate continuity (muwalat) between the two combined prayers, so if there is a separation between them, the combination becomes void. However, some scholars do not stipulate continuity. What your friend did by performing the second prayer individually is sufficient, as it is permissible to perform one of the combined prayers in congregation and the other individually.