Just to clarify, the fiancé visa IS the marriage visa ( a visa for getting married in the U.K.), so your visa path is:
Marriage (fiancé) visa -> FLR(M)
By post, it’s currently taking about 6-8 weeks for FLR(M) applications... which is pretty fast (it has taken several months in the past). It’s up to you whether she’s happy to wait that long or if she wants to be able to work sooner than that.
Someone on a fiancé visa should be fine to sign a tenancy agreement, as they have a valid visa to live in the UK. I haven’t heard of anyone having problems before.
Anyone can be added to the council tax, regardless of whether they can work or not.
Your documents look okay to me. The most recent payslip and bank statement, plus the employer letter must be dated within 28 days of either the postmarked date of the documents package if applying by post, or the in-person appointment date.
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