Why is Puerto Rico typically pronounced like "Porda Rico" by many English speakers?

Short answer: Because that's how it's pronounced.

This is the mayor of San Juan speaking in English, pronouncing "Puerto Rico" at 3:48 of the video: https://youtu.be/N453aogbfaU?t=3m45s. I'm guessing she knows how it's pronounced. And to me, it sounds a lot like the pronunciation you're denouncing.

That being said, I know there are a lot of Latinos who, in the middle of an English sentence, will suddenly shift and pronounce a Spanish name a la española, but that is just code-shifting or hypercorrection, and doesn't really indicate how a Spanish-language name is normally pronounced in English.

It seems that all of the parts of your question relate to the pronunciation of "Puerto" specifically, so I will just talk about that.

About the first vowel:

• Edit: nohat♦ made a very good point in a comment: the name actually used to be spelled as "Porto Rico" in English. The Oxford English Dictionary entry on "Puerto Rican" says

  Puerto Rico (formerly also Porto Rico), the name of an island in the Greater Antilles group of the West Indies [...] The former English form Porto Rico is probably < French Porto-Rico (late 18th cent. or earlier), which is itself an alteration (probably after French port port n.1) of Spanish Puerto Rico. Spanish †Porto Rico is only occasionally attested in the 17th cent.

  The name was officially changed in 1932 from Porto Rico to Puerto Rico. [my note: you can see this at 48 U.S. Code § 731a - Change of name; Puerto Rico] The Commonwealth of Puerto Rico, in voluntary association with the United States, was established on 25 July 1952.

• The Spanish diphthong "ue" actually comes from "o" and it seems that some Spanish speakers currently pronounce something like "o" at least some of the time in words spelled with "ue" in standard written Spanish. See the following WordReference thread: fuego, luego (pronunciation) - ue > o

• There are no common English words starting with /pwɛr/ or /pwer/. The consonant cluster /pw/ only occurs in words recently taken from a foreign language, or still felt as foreign. There are many common English words starting with /por/ or /pɔr/, like port, pork, porous. So the pronunciation starting with the "por" sound of "port" comes more naturally to an English speaker (and it doesn't hurt that the Spanish word "puerto" is actually related to the English word "port").

About the middle consonant:

• It's natural for many English speakers to pronounce the sound /t/ in a way that sounds like /d/ in this position. (This weakened or "lenited" pronunciation of /t/ is often transcribed as [ɾ]; in some learner's dictionaries, it is transcribed as "t̬" with the voicing diacritic added to the base /t/ symbol.) Pronouncing a voiceless [t] here requires an unnatural suspension of usual American English pronunciation habits.

• Even if we consider the Spanish pronunciation, Spanish speakers usually don't aspirate voiceless consonants the way English speakers do, so a [tʰ] sound like at the start of the English word "toe" or "tow" would probably not be quite "right" in terms of Spanish pronunciation anyway.

• In fact, Spanish intervocalic /p t k/ can in some accents be pronounced as partially or fully voiced plosives [b d g] (or even as voiced continuants, albeit ones with more constriction on average than underlying /b d g/ in intervocalic position). See the following document for more details: "Consonant lenition and phonological recategorization", José Ignacio Hualde, Miquel Simonet & Marianna Nadeu

About the final vowel:

• In English, the sound /o/ can often be lenited to a schwa-like sound, [ə], when it is unstressed. Consider the pronunciation of "photographic": many people say something like "phodagraphic", not "pho-tow-graphic".

I agree almost fully with the explanation "There are no common English words starting with /pwɛr/ or /pwer/. The consonant cluster /pw/ only occurs in words recently taken from a foreign language, or still felt as foreign."

English has Pueblo (Colorado), puerile, and puerility, all of which are integrated.

In addition to that explanations, there is this one:

Inasmuch as the English name Portoo Rico ~ Porto Rico ~ Porto-Rico is attested since at least 1698, it is not surprising that by now the monophthongal pronunciation has become widespread.

A correction to the OED, mentioned above: the currently earliest-known use of the Spanish form Porto Rico by native speakers of Spanish uninfluenced by usage in any other language dates to 1562 and such evidence continues down to the twentieth century at least.

Details here:

Gold, David L. 2012. "The Politicization of a Monophthong: A Refutation of All the Puerto Rican Myths About the Native Spanish Place Name Porto Rico." In Estudios de lingüística española: Homenaje a Manuel Seco. Félix Rodríguez González, ed. Alicante. Publicaciones de la Universidad de Alicante. Pp. 215-268.

With regard to the origin of "Porto Rico" in English-language texts, it may be of interest that Early English Books Online finds 50 unique sources containing the term from the period 1599–1700. The earliest of these is Richard Haykluyt, The Principal Nauigations, Voyages, Traffiques and Discoueries of the English Nation (1599–1600), where it appears four times.

First, from "A voyage with three tall ships, the Penelope Admirall, the Marchant royall Viceadmirall, and the Edward Bonauenture Rereadmirall, to the East Indies, by the Cape of Buona Speransa, to Quitangone neere Mosambique, to the Iles of Comoro and Zanzibar on the backeside of Africa, and beyond Cape Comori in India, to the Iles of Nicubar and of Gomes Polo within two leagues of Sumatra, to the Ilands of Pulo Pinaom, and thence to the maineland of Malacca, begunne by M. George Raymond, in the yeere 1591, and performed by M. Iames Lancaster, and written from the mouth of Edmund Barker of Ipswich, his lieutenant in the sayd voyage, by M. Richard Hakluyt":

Our victuals now being vtterly spent, & hauing eaten hides 6 or 7 daies, we thought it best to beare back againe for Dominica, & the Islands adioyning, knowing that there we might haue some reliefe, whereupon we turned backe for the said Islands. But before we could get thither the winde scanted vpon vs, which did greatly endanger vs for lacke of fresh water and victuals: so that we were constrained to beare vp to the Westward to certaine other Ilandes called the Nueblas or cloudie Ilands, towards the Ile of S. Iuan de porto Rico, where at our arriuall we found land-crabs and fresh water, and tortoyses, which come most on lande about the full of the moone.

From "The voiage made by Sir Richard Greenuile, for Sir Walter Ralegh, to Virginia, in the yeere 1585" (header note):

They land vpon the Iland of S. Iohn de Porto Rico.

From "A notable discourse of M. Iohn Chilton, touching the people, maners, mines, cities, riches, forces, and other memorable things of New Spaine, and other prouinces in the West Indies, seene and noted by himselfe in the time of his trauels, continued in those parts, the space of seuenteene or eighteene yeeres":

From this place [Jamaica] they go to the cape of S. Anthony, which is the vttermost part of the Westward of the Island of Cuba, and from thence to Hauana lying hard by, which is the chiefest port that the king of Spaine hath in all the countreys of the Indies, and of greatest importance: for all the ships, both from Peru, Hunduras, Porto rico, S. Domingo, Iamaica, and all other places in his Indies, arriue there in their returne to Spaine, for that in this port they take in victuals and water, and the most part of their lading: here they meet from all the foresayd places alwayes in the beginning of May by the kings commandement: at the entrance of this port it is so narrow, that there can scarse come in two ships together, although it be aboue sixe fadome deepe in the narrowest place of it.

And from "A relation of the commodities of Noua Hispania, and the maners of the inhabitants, written by Henry Hawks merchant, which liued fiue yeeres in the sayd countrey, and drew the same at the request of M. Richard Hakluyt Esquire of Eston in the county of Hereford, 1572":

There is in New Spaine a maruellous increase of cattel, which dayly do increase, and they are of a greater growth then ours are. You may haue a great steere that hath an hundred weight of tallow in his belly for sixteene shillings; and some one man hath 20000 head of cattel of his owne. They sell the hides vnto the merchants, who lade into Spaine as many as may be well spared. They spend many in the countrey in shooes and boots, and in the mines: and as the countrey is great, so is the increase of the cattell woonderfull. In the Island of Santo Domingo they commonly kill the beasts for their hides and tallow; and the fowles eat the carkaises: and so they do in Cuba and Porto Rico, whereas there is much sugar, and cana fistula, which dayly they send into Spaine.

If Hawks's relation of the lands and commodities of New Spain dates to 1572, as the title indicates, the spelling "Porto Rico" in Ebglish is at least that old. Chilton's account, although written in 1586, is based on notes he recorded during the years 1568 to 1586 and so may reflect an even earlier instance of the spelling.

On the other hand, the spelling "Puerto Rico" appears 53 times in Hakluyt's Voyages, including accounts written in English or translated into English at least as early as 1553. Indeed "Puerto Rico" also appears in English translations of books by Spanish auhors published in 1580 and 1588, as well as in Walter Raleigh's The Discouerie of the Large, Rich, and Bewtiful Empire of Guiana (1596). Overall, however, the number of unique books in EEBO's search results containing the spelling "Porto Rico" (50) is about double the number of unique books containing the spelling "Purto Rico" (24).

If nothing else, these results indicate that English readers have had a long time to be confused aboout how the island's name is spelled and (by extension) pronounced.