The Firebox
The firebox is the part you can see, and what it looks like tells you a lot about what is going on in the parts you cannot.
Short answer
Check the firebrick and the mortar joints, the damper, the throat and the smoke chamber. Deteriorated firebrick or open joints in the firebox means heat reaching the structure behind, which is a genuine fire risk. Water staining and a rusted damper means the top of the chimney is letting water in.
What to look at
Firebrick condition: cracked, spalled, missing or loose brick, which is a repair rather than a cosmetic matter.
The mortar joints between the firebrick, which should be tight — open joints let heat through to the structure behind.
The back wall and the throat, which take the most heat.
The damper: does it open fully, close properly, and is it rusted or jammed.
The smoke shelf behind the damper, for debris, mortar and nesting material.
And the smoke chamber above, which should be smooth-sided and frequently is not.
Open joints are not cosmetic
The firebox is the only thing between a fire and the combustible structure behind it. Open mortar joints and missing firebrick let heat and flame reach that structure. It is a repair rather than a tidy-up, and a sweep who flags it is not upselling.
What the damage tells you
Water staining on the firebox back or floor: water coming down the flue, which means the crown or the cap.
A rusted damper: the same, over a longer period.
Debris in the firebox: tiles or mortar failing above, which is a liner question.
White efflorescence inside the firebox: water moving through the masonry.
Heavy soot on the firebox face above the opening: smoke spilling into the room, which is a draft problem.
And cracked or heat-damaged firebrick with a history of very hot fires, which also suggests looking at the flue for creosote and for heat damage.
What cleaning involves and does not
Ash removed, with the understanding that a thin layer of ash in a wood fireplace is useful rather than a problem.
The firebox brushed down and vacuumed, including the throat and the smoke shelf.
Glass doors cleaned, and the gasket checked on a sealed appliance.
The damper cleaned and operated.
Not pressure washing or wet cleaning the firebox, which drives water into the masonry.
And not painting or coating the firebox interior with anything not rated for it, which is a thing people do and should not.
Common questions
Do open joints in the firebox matter?
Yes. They let heat reach the structure behind. It is a repair.
What does a rusted damper mean?
Water has been coming down the flue for a long time. Check the crown and the cap.
Should I remove all the ash?
A thin layer in a wood fireplace is useful. Remove the bulk of it.