Are accrued interest expenses, rent and operating leases interest bearing debt?

Interest payable is an accrued expense - this is a circular argument. If a note requires quarterly interest payments, each month will have an entry that debits interest expense and credits interest payable, recognizing the portion of the interest that accrued during the period. A loan isn't going to have interest that compounds at a rate faster than it is repaid - that is the definition of insolvency, so accrued interest isn't considered "interest bearing debt". I don't want to get into APR and EAR, but interest expense and the related liability are calculated using EAR regardless of what APR is, so compounding is still accounted for.

Accrued rent and operating leases, again, are liabilities created to recognize the portion of the expense incurred during the period. If a company is making late payments and there are fees associated, those are expenses as well - but it doesn't mean you treat the accrued portion as interest bearing debt. If the company is unable to meet these obligations and they finance the payments, THAT would be IBD.

Not sure what invested capital has to do with any of this. Accrued expenses are liabilities, so they balance against assets. A trial balance (assuming this is what you are referring to as "analytical balance sheet" - never heard of that) includes expenses (debit normal balance) which will balance against the related liability for the period (credit normal balance). Invested capital balances against stockholders equity, as well as any assets that were purchased using that capital.