Which witch is which?

Although 'plain' as in Answer 3 is a flat formation, the tool used by joiners to achieve a similar smooth effect is a Plane (this word means a flat surface in geometry, hence the more-or-less flat wing surfaces of an aircraft, e.g. the tail-planes, and indeed also the hydroplanes of a submarine which do a somewhat comparable job).
Metaphorically we also sometimes refer to something being 'on a higher plane', e.g. that Calculus is rather 'above' most people whose Maths only reached about GCSE standard.
'Plane' (spelt thus) is also the name of the species of tree fairly widely seen in London and other city streets.

Yes: little old 'or' is a conjunction; the rest are all nouns.

'Right' can also be a noun (as in 'human rights') and an adjective ('the right hand'), even a verb ('you need to right a capsized vessel as soon as possible'); but here it is an adverb, describing the manner of doing an action.

'Sore' is an adjective, used here metaphorically of course.
'Saw' in Answer 1 is the plain past form of 'see', but in Answer 2 it is the present form of 'to saw' ( = to cut with a saw, as a carpenter would do).
'Soaring' (Answer 4) sounds practically identical but is in no sense related to any of the previous words.

There is (so far as we know) no such thing as a 'lert' (Answer 4); so this is one of those rather silly puns that ought to work, but the joke ~ such as it is ~ is that it doesn't.

'Rowed' in Answer 1 is the past form of the verb 'to row' (i.e. to propel a boat, using oars ~ see Q.2 A.4 above!); in Answer 3 it is the past form of 'to row' as in 'to make an embarrassing and noisy fuss about something' (here, it rhymes with 'how').
Your author was once somewhat disconcerted to attend a happy wedding where one set of seats was marked with a sign, 'FAMILY ROW'. Obviously in this context it referred to a 'row' of seats side-by-side (rhymes with 'no'), rather than indicating a major argument 'to rhyme with 'now'), despite the identical spelling.
The chicken joke (in Answer 4), and numerous variations, is a standard English verbal institution. The pun here is on the American placename, Rhode Island (sounding like a 'road island', i.e. a little raised refuge halfway across a main street, where pedestrians crossing can pause before tackling the second stream of traffic).

The 'lead' (or sometimes 'leash'), along with the present form of its related verb, is pronounced with a long E; all the others are short, despite the '-ea-' spelling in the chemical element Lead (= 'Pb' to chemists, from the same Latin root that we have Plumber, originally someone who worked with water-piping that was made of lead. There are British-made water-pipes in the ruins of Herculaneum, apparently ... how about that for an example of the European market under the Romans, 2,000-odd years ago?)

In Answer 2, 'weight' and 'wait' are reversed.
In Answer 4 there is the further echo of a rather poor pun on 'a frayed knot' (i.e. one in which the ropes themselves are worn and 'frayed', making it even harder to untie cleanly or quickly).

'Need' can certainly be a noun as well as a verb ('A friend in need is a friend indeed' ... or, as some cynic once commented, viewing the friendship from its other end and a rather less charitable perspective, 'A friend in need is a ~-ing nuisance'.)
Meanwhile 'knee' is usually a noun, but there is a verb 'to knee', and that's what is called for in Answer 3. One almost feels it should have a third 'e' in it, but two are enough (cf. 'freed, agreed' etc.).

'Scene' and 'seen' are reversed in Answer 1. A good poet would (or should) certainly think twice before writing anything as weak as:
'It was a calm and tranquil scene. / The prettiest I'd ever seen.'
(That somehow goes beyond the idea of rhyme and comes out the far side, not having satisfied!)
We hope you will have been making good use of a substantial dictionary to disentangle any other rare, unfamiliar or otherwise potentially troublesome usages. And please don't now get us started on 'red' and 'read' (in its past tense/sense)!