The origin of the term frog in relation to the braid on military uniforms (and almost uniquely cavalry uniforms) might come from the old term for frock. The OED has:
Frock 2. An upper garment worn chiefly by men... 1375. Barbour Bruce x. 375 With blak froggis all helit thai The Armouris at thai on thame had. c 1425 Wyntoun Cron. viii. xxxviii. 57 Ilkane a gud Burdowne in hand, And royd Frogis on þare Armyng. c 1460 Towneley Myst. (Surtees) 241, I wold be fayn of this frog [Christ's coat] myght it fall vnto me. 1500–20 Dunbar Poems li. 3 To giff a doublett he is als doure, As it war off ane futt syd frog
A 14th Century 'frock' or 'frog'. Note the belt and cord carrying gear. The term frog-loop and frog-belt have been used to describe these items up to the 19th Century (see below) and even today a frog-loop is still a term for a retainer for a tool in a tradesman's belt.
There are also references in the Dictionary of the Scots Language:
Frog, Froig, n.1 [ME. frog, frogge (15th c.), of uncertain origin.] A frock; a cloak or coat. Barb. x. 375.
With blak froggis all helyt thai The armouris at thai on thame had;
Wynt. viii. 5702.
[They had] royd frogis on thare armyng, To covyre thame for persaywyng;
Crying of Play 39.
Five thousand ellis ȝeid in his frog Of hieland pladdis of haire;
Dunb. li. 3.
To giff a doublett he is als doure, As it war off ane futt syd frog;
Doug. vi. v. 132.
In hevy wayt frog stad and chargyt soyr;
1550–1 Treas. Acc. IX. 467.
Spanȝe freis … to be ane froig to my Lady;
So what we might be looking at is frog(frock)-button, and frog(frock)-belt, as below:
1719 De Foe Crusoe ii. iv. (1840) II. 68 He drew a hatchet out of a frog-belt. 1827 Hone Every-day Bk. II. 190 A coat with frog-buttons. 1867 Smyth Sailor's Word-bk., Frog-belt, a baldrick.
Curiously (and perhaps destructively to this argument) Defoe 1719 also has:
1719 De Foe Crusoe i. xv, A belt with a frog hanging to it, such as..we wear hangers in.
Tailoring of simple jackets, frocks or 'frogs' only became usual from the 1500's onwards. Prior to that they were held in shape with belts, cords, lacing and over the shoulder straps which served the dual function of carrying tools and possessions on hooks and loops, and in bags hung from them. Pockets are a relatively late invention in clothing. (See: http://www.theinnerbailey.com/baileybasics-sets-townsman-1.jpg)
My suggestion (tending towards an answer) is that this early fashion of belting, lacing and buttoning was contemporary with the use of the ME word 'frog' for jacket, and the the technical expressions for items associated with them such as frog-button, frog-belt, frog-loop and frogging were all developed during that time. As fashion evolved in the 1500-1600 period and clothes were more tailored to the shape of the body and pockets came into use, not only did the term 'frog' evolve into 'frock', but those belts and cords became less important for shaping the material and for carrying items. Frog as a word for a frock evolved, but the use of frog in association with those accoutrements became fossilized.
To see how these fossilized terms for buttoning and braiding (frog-buttons, frogging) came to be associated with a military fashion of the early 1700's (in England) we need to look at Eastern Europe in the late 1400's.
Frogging (as we call it) is most closely, and most extravagantly, associated with lightly armoured mounted troops known as Hussars or Dragoons. This link to an image from c.1550 clearly shows the horizontal braid on a mounted Hungarian soldier's uniform:
Such troops, and their dress styles, evolved in Eastern Europe before spreading to Western Europe in the late 1600's. None of the languages or cultures (including that of the Ottoman Empire) appear to have contributed the word 'frog' or 'frogging' along the way.
The actual fashion which became quickly extravagant and an actual impediment to military action (1746 Berkeley Let. Wks. 1871 IV. 306 Laces, frogs, cockades..are so many..obstacles to a soldier's exerting his strength.) may have originated in loops in the jacket to hold additional weapons such as daggers and short swords, or in the lacing used to tighten loose plates of armour or a baggy cloth jacket around the body.
When the new style reached England in the mid 1700's there was no current word to describe the braiding (that had become increasingly ornate as it travelled west across Europe). It was recognized, however, as similar to the ornamentation formerly (and no longer) used on the item of clothing once known as a frog. The names for that ornamentation (including 'frogging') had never evolved because the items had been largely disused for about a hundred years, but were revived in their original form to describe the ornamentation on this new (to English eyes) military fashion.
This speculative essay into the evolution of the use of the word 'frog', and the suggestion that it's original association with the article of clothing, and the accoutrements to that clothing went down different paths, and that the word for the accoutrements was frozen for some one or two hundred years while the word for the clothing evolved, and how the frozen word was revived to describe a new fashion element, also explains how we arrived at DeFoes (otherwise improbable) description of a 'frock with frog-buttons'. Essentially he is saying, 'a frock with frock-buttons', but the language available to him has him say otherwise.
'Dandelion' is, as you correctly pointed out, from the French, 'dent-de-lioun'.
Etymonline shows:
early 15c., earlier dent-de-lioun (late 14c.), from Middle French dent de lion, literally "lion's tooth" (from its toothed leaves), translation of Medieval Latin dens leonis.
Other folk names, like tell-time refer to the custom of telling the time by blowing the white seed (the number of puffs required to blow them all off supposedly being the number of the hour), or to the plant's more authentic diuretic qualities, preserved in Middle English piss-a-bed and French pissenlit.
Looking into 'pissenlit' on a French etymology dictionary reveals:
BOT. Plante de la famille des Composées, vivace, à feuilles longues dentelées disposées en rosette, à fleurs jaunes et à petits fruits secs surmontés d'une aigrette. Synon. dent-de-lion (s.v. dent). Pissenlit officinal; feuille, fleur, graine, racine de pissenlit; salade de pissenlit; culture du pissenlit. Les feuilles radicales longues et couchées du pissenlit sont dentées, en lobes arqués et renversés, glabres et d'un beau vert. [...]
Comp. de pisse (forme du verbe pisser*), de la prép. en* et de lit*. Cette plante est ainsi nommée en raison de ses propriétés diurétiques. Fréq. abs. littér.: 77. Bbg. Schurter (H.). −Die Ausdrücke für den Löwenzahn im Gallo-romanischen. Halle, 1921 pp.8-24, 89-90, 105, 106, 110.
Which roughly translates to:
BOT. plant of the composite family, perennial, with long serrated leaves arranged in a rosette, yellow flowers and small dried fruit with a Pappus. Synon. Dandelion (s.v. tooth). Officinal dandelion; leaf, flower, seed, dandelion root; Dandelion salad; culture of the dandelion. The radical long, folded the dandelion leaves are toothed, lobes, arched and overturned, glabrous and green. [...]
Comp. piss (piss the verb form *), prep. in * and bed *. This plant is so named due to its diuretic properties. Freq. ABS. litter.: 77. BBG Schurter (H). −Die expressions as den Lowenzahn im Gallo-Roman. Halle, 1921 pp.8 - 24, 89-90, 105, 106, 110.
So apparently 'dent-de-lion' is synonymous with 'pissenlit', but didn't make mainstream usage, while England went the other way - apparently 'piss-a-bed' used to be another term for it, as Oxford shows:
Dandelion
Late 16th century: from the verb piss (because of its diuretic properties) + abed, suggested by the French name for the dandelion, pissenlit.
but we preferred to use 'dandelion' instead.1
Ultimately, both 'pissabed' and 'dandelion' are English words for the flower, and 'dent-de-lion' and 'pissenlit' are French for it as well, but the popularity seems to be swapped round, with English using dandelion and French using pissenlit.
This may be due to other European nations such as Italy using related etymology (pisacan).
1: Just speculating, but this could have been due to French-speaking nobility being in power, such as Henry VII and Henry VIII, and the French terms filtering down (like the difference between 'beef' and 'cow'). I'm not too sure about how language worked historically in France, and so can't offer any help as to why the French chose 'pissenlit' rather than 'dent-de-lion', possibly because of confusion with real lion's teeth?
editing in progress...
Best Answer
The 1921 work, Dogs of China & Japan in Nature and Art, by V.W.F. Collier, cited in OED (paywalled) as the origin of the English name ('shih-tzu') of the breed, completely addresses your question, although not directly:
Collier adopted 'Shih-tzu' (not shih-tzu kou) as the name of the breed in English.
The chronological history of the later (?) adoption of Xi Shi in contemporary Chinese as the name of the breed, and (presumably) its subsequent adoption and use in English as an alternative name of the shi-tzu breed, cannot be readily ascertained by this non-Chinese speaker.