Is it considered a shortcoming if a person is unable to do good or understand religion due to intellectual, financial, or health limitations? And will he be held accountable for not doing what he intended to do?

Accountability applies to the abandonment of an obligation or the commission of a forbidden act. The examples mentioned are desirable matters for which there is no accountability. Intention holds great significance, for whoever sincerely intends an act of worship, and is then prevented from performing it by a reason beyond their control, will receive the reward of those who performed it. Allah the Almighty said: "And whoever leaves his home as an emigrant to Allah and His Messenger and then death overtakes him—his reward has already become incumbent upon Allah." The Prophet, peace and blessings be upon him, said: "Whoever sincerely asks for martyrdom, Allah will grant him its reward, even if he does not attain it," and "If a servant falls ill or travels, the like of what he used to do when he was healthy and resident will be written for him." So, whoever intends good but is unable to perform it due to illness or another reason, the reward for what he intended will be written for him, if his intention was sincere.