There is no such word as "dis". It is an extremely informal style. Really, it's just showing laziness. Don't use "dis" in place of "this" unless you're trying to be funny. Dis dude deserves a dunking.

In American accent, unlike UK accent, we do not have ejectives on our T's, which basically means we say the t sound before we push out our breath. You know that sound that a hihat makes? That t-t-t-t-t sound. That's what we DON'T do. So "medal" sounds the same as "metal" much to the dismay of our British friends. So because basically t is the same as d, this is shortened to dis irregardless that "Th" isn't 'T" It's just a way to type faster.

You're right, 'd' and 'th' are different sounds, but 'd' is probably the closest sound in English to 'th'. When English speakers are drunk or extremely tired, they'll sometimes pronounce 'th' as 'd'. It's also pronounced like this in some accents in the southern US. It is considered improper English, but some people will write it in humorous imitation of either of these.

Dis the verb and dis the noun are actually officially slang words in American English and appear in a number of dictionaries. See the links at the end of my response to see their definitions on their corresponding online dictionary websites. "Dis" is also slang for "this". For instance, if two women are fighting on the subway you might hear "Can you believe dis b*tch just called me a ho?!?" It doesn't appear in the dictionary yet but it is in use in very informal contexts in America. If you use it, people may think you are uneducated. Hehehehe. Anyhow, you are right. The d sound doesn't sound like a th, but that is how people say it and also how they write it. This might be because it is easier to say a d than a th or because the word came out of the Bronx or some other area where people had a thick accent. I am not 100% sure but that is the reality of it though. "Dis" can also be used as a verb in slang. In means "to disrespect someone." For instance, if you call someone a jerk or say that they couldn't sing if their life depended on it, they might say "I can not believe he dissed me like that!" I remember when I was a kid, a lot of people, including my friends used to use this form of the word a lot. "Dis" is also a noun and means disrespect or a disrespectful remark. For instance, "Every one thought what he said was a dis." Again, we used to use this word a lot when I was a kid too. :) By the way, "dis" can also be spelled "diss", although, I usually always see it spelled as "dis".

See the following links for the verb and noun definitions: 1. http://www.merriam-webster.com/dictionary/dis 2. http://www.thefreedictionary.com/dis 3. http://dictionary.reference.com/browse/dis?s=t 4. http://www.oxforddictionaries.com/us/definition/american_english/dis?q=dis (Even has a few example uses) 5. http://www.collinsdictionary.com/dictionary/american/dis?showCookiePolicy=true 6. http://www.collinsdictionary.com/dictionary/english/dis?showCookiePolicy=true (listed under the British English section)