Why is Puerto Rico typically pronounced like "Porda Rico" by many English speakers?
Solution 1:
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:
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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".
Solution 2:
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.