yes
not necessarily.
for most purposes, '
inner london' is (with minor tinkering at the edges) the area of the 1889 london county council - the area north of the river is much smaller than south of the river, covering the current boroughs of (from the west) hammersmith + fulham, kensington + chelsea, westminster, camden, islington, hackney, tower hamlets.
before 1889, all of these except the 'square mile' city of london were in middlesex (south london was either surrey or kent.)
l
today's haringey borough was in middlesex until 1965, and places east of the river lea (canning town, stratford and so on) were in essex.
south of the river, the LCC area extended out to include current lambeth, southwark, lewisham and greenwich boroughs, so goes out to places like plumstead, abbey wood and crystal palace.
i'm sure it all made sense in 1889