A Letter and Puzzle From Grandma

Question

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[ EDIT: It's OK to skip the long, boring letter for now, and go straight to Stage 1 of the puzzle.]

Dear Grandson,

I could use a break, so I came to your grandpa's farm to unwind. However, woe to me! Animal sounds here seem to never stop- a dove coo, a cow moo, a goat maa, a crow caw, a wee kitten mew, auk cries, a yak grunt, ayu splashing, emu drumming.... I even wanted to sic Milo on those emus! Speaking of that goat- he always looks sus, as if he plans to break into my veggie garden. Even apart from animal noises, last night we were jolted awake by thunder. I made an eek sound, while Grandpa said aue!

Wow, sorry I'm being negative. [Switching my outlook] I am retiring in May- time to do my world traveling! I'm typing this under my favorite acacia koa tree with a sea view- a small cay is even visible on clear days. I'm eating a homegrown yam- mmm! Also, we started stir frying in a wok, which makes veggies addictive. I relax by repeating fifty oms each morning while focusing on dissolving my ego. Is it working? Aye, of course, better than a gym workout! Grandpa agrees, so... ays have it!

About a week ago, Mr. Smith visited us. He is quite a sui generis gentleman, seemingly an expert in every field! He used to be an ace MiG test pilot. We love to just ask questions to hear amazing answers. First, he explained to us what yaw means. Then he went into tactics he used in a countersuit, sui juris, in response to Dunn County trying to sue over a property boundary issue. Next, he spoke of gem mining, touched on wig making, expounded some benefits of yag laser surgery, clarified cam replacement on Grandpa's truck, provided tips on saving gas, revealed tricks that coy gents might apply to woo their soulmates, showed us karate championship photos with everyone posed in white gis, displayed gay pride parade photos, demonstrated escaping a wolf's maw, ... among other things that I don't recall, I'm sure.

Anyway, I know I owe you a puzzle. It is attached. Your goal is to name every country I will visit when I retire.

With Love, Grandma

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Partial answers are fine if you are adding at least one stage's answer that was not yet answered.

Stage 1: Find these 5 non-overlapping paths.

Tip: Look off-path.

[Edit: Each number represents a move in the path.]

Paths are horizontal and vertical only.

Path #1: 7,1,20,14,3,8,18,19,1,14,2,7,8,4,5,9,2,8,19,17,12,3,2,3,6,3,15,1,6
Path #2: 15,14,7,2,6,8,3,11,3,16,9,2,5,7,19,5,1,4,18,3
Path #3: 14,9,6,3,4,9,9,1,7,5,1,5,12,17,1,2,15,16,1,12,6
Path #4: 10,18,11,10,7,4,14,5
Path #5: 16,6,16,14

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Text version of the grid above:
EAOTACEBHE
BTPSHFRADT
EHNEMNLTJU
IORBKWRVIO
RDAILSUNST
IFMONEFYAL
JTERHROGIB
CHIUTDSNES
OBNEYXTCIN
ASRTRDESYJ

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Stage 2: Apply the Stage 1 clue.

Tip: Apply the clue (Horizontal, vertical, and diagonal) to the grid and ignore Grandma's letter (and countries) until stage 3.

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Stage 3: Apply the Stage 2 clue.

Tip: Think equidistant.

[Edit] Added in a tip above.

[Edit #2] Not equidistant on a map (geographically), but in another way.

[Edit #3] Stage 3's "equidistant" is in alignment with its meaning in the puzzle's theme.

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Stage 4: It's research time. (easy, don't worry)

Could be considered a part of stage 3, but I separate them here.

Tip: knowledge

Answer 1

Stage 1

The puzzle mentions that "paths are horizontal and vertical only" and that "each number represents a move in the path". It would follow that each number dictates, in some way, the relationship between the two letters the path goes between. Specifically, the number is the difference between the numerical equivalents of those two letters. We can then logically and uniquely draw out each of the five paths as so:

Reading out the unused letters gives the cluephrase ADJ LTRS NOT ADJ, meaning ADJACENT LETTERS NOT ADJACENT.

Stage 2

To solve stage 2, you need the cluephrase from stage 1, ADJACENT LETTERS NOT ADJACENT, and the grid of letters. If you cross out every pair of adjacent letters (including diagonal) in the grid that are also adjacent in the alphabet, you get the following grid:

Reading out the remaining letters gives EACH THREE LTR WORD IN MY LTR HIDES ONE CNTRY, which means EACH THREE LETTER WORD IN MY LETTER HIDES ONE COUNTRY.

Stage 3

This stage uses the cluephrase EACH THREE LETTER WORD IN MY LETTER HIDES ONE COUNTRY. In fact, there are 50 three-letter words that can be extracted from the letter. The most interesting aspect of the words is that each of the letters has an odd position in the alphabet. This, combined with the "equidistant" tip, clues that you need to take the letter which is alphabetically in between each pair of letters. You can do this with both pairs of adjacent letters within the trigram to get a list of bigrams. For example, USE becomes TL because T is halfway between U and S, and L is halfway between S and E.

Stage 4

Finally, we have to identify a list of countries. As it happens, these bigrams from stage 3 all happen to be ISO 2-letter country codes, meaning each bigram can be transformed into a country name. The resulting bigrams and country names can be seen below:
USE TL Timor-Leste
WOE SJ Svalbard and Jan Mayen
COO IO British Indian Ocean Territory
COW IS Iceland
MOO NO Norway
MAA GA Gabon
CAW BL Saint Barthélemy
WEE NE Niger
MEW IN India
AUK KP Korea (Democratic People's Republic of)
YAK MF Saint Martin (French part)
AYU MW Malawi
EMU IQ Iraq
SIC NF Norfolk Island
SUS TT Trinidad and Tobago
EEK EH Western Sahara
AUE KM Comoros
WOW SS South Sudan
MAY GM Gambia
KOA MH Marshall Islands
SEA LC Saint Lucia
CAY BM Bermuda
YAM MG Madagascar
MMM MM Myanmar
WOK SM San Marino
OMS NP Nepal
EGO FK Falkland Islands (Malvinas)
AYE MO Macao
GYM PS Palestine, State of
AYS MV Maldives
AGO DK Denmark
SUI TO Tonga
ACE BD Bangladesh
MIG KH Cambodia
ASK JO Jordan
YAW ML Mali
SUI TO Tonga
SUE TM Turkmenistan
GEM FI Finland
WIG PH Philippines
YAG MD Moldova, Republic of
CAM BG Bulgaria
GAS DJ Djibouti
COY IT Italy
WOO SO Somalia
GIS HN Honduras
GAY DM Dominica
MAW GL Greenland
OWE SN Senegal
YOU TR Türkiye