First of all, welcome to the forum.
F4 was a 40 guinea gun built on the 1880 patent Facile Princeps action with Greener's 'treble wedge fast' locking system. Your gun was made in 1900.
I cannot see any nitro proof marks on your gun from the photos, though the proof marks suggest it was proofed under 1896-1904 rules in Birmingham. It is a nominal 12-bore with tight bores (hence the 13 stamp). Chambers are 2 1/2".
This is a nice quality, well made gun.
Removal of the dent will depend on wall thickness and the skill of the man you engage to raise it. It looks quite sharp: a 'soft' dent can be raised with less residual effect.
The barrel steel refers to the fluid pressed forging system devised by Sir Joseph Whitworth, it heralded the replacement of damascus steel as a gun barrel material.
