Word-Number Codes 1

PINT and NEST both end with T, and 1467 and 6237 both end with 7, so we know those two codes match those two words. Next we need to look for other code/letter matches. 1467 and 3146 both have a 1 (in first and second position) so we need words which follow that pattern. The letter P works. So: PINT = 1467 and SPIN = 3146. Since we know that N = 6, 6237 must = NEST.
Knowing all this allows us to say that SNIP = 3641

We are aiming for a word starting with G, and among our given words there were two that also began with this ~ and one that ended in it. No other letter is so conspicuous except, perhaps the D on the back of POND and GOLD (a pattern of -O-D, which might also catch our eye if we happen to have been granted the codes for both of them), or possibly the P's in GRIP and POND. We have to start somewhere!
Taking 8163 first, there's no sign of the initial 8 on the front or back of anything else, so that must probably stand for F, in which case we have FLAG, and 3 = G. 3205 must either be GRIP or GOLD; but it doesn't 'rhyme' with 5974, which must be POND, while GOLD must have been 3X14 or something similar.
GIRL must start with 3, followed by the I (which must be 0, as in 3205 = GRIP) and the R after it, instead of before: hence 302X; and we can check that L = 1 from back in FLAG. So GIRL must be 3021 after all!

We do not seem to have a code given for HIGH, which would have the same number on each end to represent the H.
Two of the other words have the I vowel in them, in the same place (3rd position), which suggests that the groups 9021 and 4726 must stand for HAIL and GRIN (or possibly, the other way round). DOGS has a G in 3rd place (as did HIGH, for which we don't have the code), so DOGS must be 8395 and 9 therefore stands for G, so 9021 must be GRIN.
LASH starts with the end of HAIL (=6); A from 2nd place in HAIL (=7) ... and the only available Answer starting this way is No.2

RUDE, DALE and TAPE all end with E, and 1653 and 5423 both end with 3, so we know those two codes match two of those three words. Next we need to look for other code/letter matches. 1653 and 5423 both have a 5 (in first and third position) so we need words which follow that pattern. The letter D works. So: DALE = 5423 and RUDE = 1653. Since we know that R = 1, 9761 must have an R in fourth position. POUR = 9761. Knowing all this allows us to say that ROPE = 1793

The most frequent letter this time is R, so it may crop up in at least two, or even all three, of our coded versions. Moreover, it begins ROAD and ends PAIR; no other letter does two jobs like this in the current group apart from the P in CHIP / PAIR, so it might be 6.
CHIP and PAIR also have an I in 3rd position each, so it looks as though P = 6 and I = 8; DROP (which we're looking for) will therefore end in a 6, narrowing our potential Answers to Nos. 2 & 4.
We do not have a number value for the letter O in DROP because we can't yet tell whether the other given number code is for ROAD or DARK, but since we know that 6281 = PAIR, 2 must be A; its comes in 2nd place in 5291 which must therefore be DARK, and ROAD would be 1X25 (three-quarters of it being 'dark' re-cycled; we've left an X for the still-unclear code for O).
This gets us close enough to know that DROP must be 51X6, hence Answer 4 can be the only right solution

We might have looked for two coded words ending in the same number, like the words ROPE and PALE each ending in -E; unfortunately this word-shape clue is not available. However, we know that HALF and PALE each have an L in 3rd place, so these must be 6857 and 1853 (or vice versa); and if the other code (7492) can't stand for ROPE, it must be FOUR. We need its final R and U to start RULE, which gives us 29XX (with two blanks, as yet, on the end); 2953 is the only one offered that would fit

LAST and MEAT both end with T, and 8647 and 5437 both end with 7, so we know those two codes match those two words. Next we need to look for other code/letter matches. 8647 and 5437 both have a 4 (in second and third position) so we need words which follow that pattern. The letter A works. So: LAST = 5437 and MEAT = 8647. Since we know that M = 8, 1458 must be PALM, So: LAST = 5437, MEAT = 8647 and PALM= 1458 Knowing all this allows us to say that TAPE = 7416

The E wanders about in three of the words (one of which we may not even have the code for), so, for such a useful and common letter, it may not offer us the handiest key this time. The next most frequent letter in usual English is T, which comes on the back of WEST and the front of TRUE; and there's no other letter in this present group that starts one word and ends another. Could there be a number that fits such placings? It looks as though 7 could be code for T, and by complete coincidence it even looks a bit similar.
In this case 1687 would be WEST and 7236 would be TRUE; 8 must be S (looking somewhat similar again, as it happens; like 7 and T), in which case 8139 must be SWUM. We still have no code for F, but we can now do -USE which must be X386, making Answer 3 the only possible correct one

We were given two words each starting with W (WAIT and WORK) but we do not have two corresponding number groups beginning with the same figure; so the front W will not offer us a way in on this occasion.
However, two of the words that we do have must be KITE and TEAM, with a telltale TE digraph in each, and we have a '29' pair that must correspond with these, so 7346 must either be WAIT or WORK. If it were WAIT we would already expect a 2 at the end, so it must be WORK instead; 6 = K, and TAKE must be 2X69. We also know A = 1 from TEAM, so the only fully right Answer could be 2169

It would be marvellous, of course, if we knew which of the codes stood for CUBE in 'clear'; but we don't, and it might not even be in the code list at all. Let's start from the frontmost (alphabetically) of the letters we know to expect, which is B: two of the words must start with this, and CUBE (if it's shown) would have the same letter in its 3rd position.
We've been given 4675 and 4568, so 4 must stand for B and 1943 must be CUBE, so CLUE must be 1X93; L must be 5 as in the 2nd of BLOT and the back of BOIL), so the whole word comes out as 1593