All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
Being independent of people and abstaining from asking them for help is encouraged and praiseworthy in Islam. It is better for the Muslim to dispense with people and refrain from asking them for help as much as he can. The Prophet said, "A man would persist in begging people to give him charity until he meets Allaah (on the Day of Resurrection) and his face will be without a shred of flesh." [Muslim] He also said, "He who asks others to give to him in charity in order to increase his own wealth is akin to one who asks for live coals. He who wishes to have more, let him have it, and he who wishes to have less, let him have it." [Muslim]
An-Nawawi wrote, "This applies to the one who asks others for charity while he is in no need of it; it is prohibited for such a person to ask people for charity. The hadeeth reads, '...in order to increase his own wealth,' and this indicates that the person does not need the charity that he is asking people for."
However, there is no harm in asking one's brother, colleague, or spouse to carry out some regular tasks on his behalf within his ability like in the examples that you mentioned in the question. However, it is better for the person to dispense with people and tend to his needs on his own. Shaykhul-Islam Ibn Taymiyyah wrote, "As for the things within the ability of people, it is permissible for the person to seek the help of others occasionally and not all the time; asking the created being for help may be lawful in some situations and prohibited in others..." [Majmooʻ Al-Fataawa]
There is no harm in asking your non-Muslim colleague to do your work when you leave for prayer and at other times if your employer allows it. The scholars stated that it is allowed to appoint the non-Muslim to carry out tasks on behalf of the Muslim with regards to the regular matters that he is allowed to carry out for himself. The Maaliki scholar Ibn Juzayy wrote:
"Whoever is eligible to manage his own affairs on his own can be appointed to manage someone else's affairs on his behalf; however, it is impermissible to appoint someone to act on behalf of his enemy (or to manage his affairs) or for a non-Muslim to act on behalf of a Muslim in a sale or Salam contract lest he would engage in prohibited transactions. It is also impermissible to appoint the non-Muslim to collect money from Muslims so that he would not be empowered over them ..." [Al-Qawaaneen Al-Fiqhiyyah]
Allaah knows best.