Is it permissible to wipe over one's shoes after Fajr prayer, having put them on while in a state of ritual purity, then Dhuhr prayer comes while one is at work, or is it obligatory for me to perform a new ablution?

It is permissible to wipe over the sandals if they were put on while in a state of purity, whether they are worn with or without socks. There is no need to remove them and wash the feet for ablution. It is also permissible to pray with the same ablution as long as it has not been invalidated.