There are a number of issues here. The question mentions:
Taxes, which I presume must be authorized and regulated by the US Constitution, but I don't know the details
Not exactly. The states existed before the Federal government. They are not created by the Federal Constitution, nor authorized by it. A number of restrictions on state powers and actions are specified by the Federal Constitution, and a number of others are imposed by Federal law. (the Federal courts have found implied restrictions beyond the explicitly stated ones.) But there is no Federal provision granting states the power to impose taxes, only restrictions on that pre-existing power.
States cannot impose taxes so as to violate rights federally guaranteed, or to place unreasonable burdens on the exercise of those rights. For example, states cannot impose different taxes or tax rates on a racial basis. States cannot impose different taxes on residents of other states temporarily present in, or doing business in the state. States cannot impose different taxes on people newly moved there from other states, compared to long-established residents. State taxes must not violate the Equal Protection clause. However, states may choose the type and amount of taxes to impose. They can use sales tax, VAT tax, property tax, income tax, excise tax, flat tax, or any combination that their legislatures pass. Different taxes may be imposed on different professions or kinds of businesses.
Does the US Constitution guarantee all citizens have the natural right to conduct their own business affairs?
Not as such, no. The Due Process and Equal Protection clauses limit to some extent the ability of a state to prohibit a particular business on a whim. But when a state asserts that a particular business is harmful, and demonstrates a plausible basis for that view, so that the law passes "Rational basis" review, the state can prohibit it, or heavily regulate it, or license it.
If so, does a citizen lose the right to legally own and operate a business if they cannot afford requisite state or local business license fees?
A state may require a license to engage in a particular occupation, and may require a fee, one-time or recurring, high or low, for that license. In addition, a tax may be imposed on those in a particular business or profession, which is not imposed on other kinds of business. For example, in many states, lawyers must pay an annual license fee, or they are not allowed to practice. So must many other regulated professions, such as hairstylist. One who cannot afford the fee may not engage in the business or profession. The state may waive or reduce fees for those too poor to afford them, but need not do so, and many states do not so so.
Similarly, the state may charge a fee for a driver's license, and one who cannot pay it may not legally drive.
Likewise, does a citizen lose the right to utilize the court system to petition for a redress of grievances, if they cannot afford the requisite court fees?
Many states have provisions waiving or lowering court fees for those who cannot afford them, but in most cases this is applied only in severe cases, say where a person would have to go without food to afford court fees. There have been a few federal cases requiring fee waivers for those who cannot afford court fees, mostly in connection with criminal defendants. There is not currently a general federal rule requiring court access for those who cannot afford court fees. Perhaps there should be. A case could be made that Equal Protection requires this, but Federal Courts have not so held.
Federal courts have held that holding people in jail or prison because they truly cannot afford fines, bail, or court fees is an unconstitutional denial of Equal Protection. But states need not waive such fees; they can be deferred and charged should the person earn enough money to (just barely) afford them. Even this rule is not yet invariably enforced, and many state courts routinely ignore it.
By the way "petition for a redress of grievances" doe snot normally refer to bringing a court case, but to asking a legislature to change a law, or asking an administrator or executive to exercise permitted discretion in a particular way.
And lastly, if a citizen is convicted of a crime or infraction, and the sentence requires the convict to utilize government services (e.g. prison services, probation services, registration services, etc.); under the US Constitution, can state government agencies providing these services legally require the convict to pay fees for these services (e.g. prison service fees, probation service fees, registration service fees), if these fees were not explicitly included in the sentence as fines?
Yes it can impose such fees, but usually only when neither the convict nor his or her dependents will be impoverished by such fees, as I understand it.
If a state attempts to pass or enforce state legislation dictating such fees, should this legislation generally be struck down as unconstitutional?
Such laws will not be held unconstitutional by US Federal courts under the Federal Constitution, unless they are found to violate Equal Protection, Due Process, or other specifically imposed restrictions on the state. For example, fees which were in practice imposed on people of one religion, but not those of another, would be struck down. But a fee imposed on everyone will not usually be overturned.
"The law in its majestic equality forbids the rich as well as the poor to steal bread from shops, beg in the streets, and sleep under bridges." -Anatole France