Meet Bob. Bob owes Alice money. Alice hires ACME debt collectors Inc. To pursue Bob for his debt to her. Bob receives letters from Acme and checks on the register of the FCA to find they are not authorized. How does this change Bob's position?
This remains unchanged. The debt is owed to Alice and ACME are merely acting as Alice's agent. The fact that ACME may or may not be acting unlawfully does not change the nature of the underlying debt.
This very much depends on the nature of the debt. The question appears to assume that ACME is required to be FCA authorised, but this is not necessarily the case, and probably isn't the case for many types of debt collectors that an average consumer is likely to hear from.
The relevant rule can be found at Article 39F which provides that debt collecting is a regulated activity if it involves a credit agreement, a relevant article 36H agreement, or a consumer hire agreement. There are also various exclusions found at Articles 39H - 39L.
Simple debts (e.g. you've not paid an invoice for something you bought) are unlikely to be within scope. See this Financial Ombudsman Service page for an explanation of the types of debt which are and are not applicable.
Comment: "but don't debt collection companies often buy the debts from original creditors?"
Yes, but then we're not talking about Alice's debt anymore. The debt is now owed to ACME. In such cases it's quite likely that the debt collection activity is exempt. See for example Article 39H(1) and 39H(5):
(1) There are excluded from articles 39D(1), 39E(1) and 39F(1) activities carried on by a person who is [...] (b) the supplier in relation to that agreement, [...]
(5) In paragraph (1), “supplier”, in relation to an agreement, means — (a) a person, other than the lender, whose transaction with the borrower is, or is to be, financed by the agreement, or (b) a person to whom the rights and duties of a person falling within sub-paragraph (a) have been passed by assignment or operation of law.
The general principle here is that collecting your own debt is not a regulated activity. Note however that the underlying agreement itself might still be regulated. For example, if you are collecting debt owed on a mortgage then even if you don't need to be regulated for the debt collection, you may still need to be for the mortgage.