What is the ruling on giving to a beggar with the intention of mercy, not with the intention of "for the sake of Allah"? And is admiring one's prayer when a person passes by considered showing off (riya'), and is there a sin entailed by that despite repeated seeking of refuge (from Allah)?

Giving money to the poor for their needs is considered charity, and the giver is rewarded for it. As for the feeling of wonder when people see you performing acts of obedience, if it is merely a thought cast by Satan, it does not harm, and you must strive to repel it, lest it lead you to blameworthy ostentation.