does sound th as d makes any difference in fast speech?

In some Irish, Liverpool and West Indian dialects the voiced dental fricative (as in 'that') can sound like 'd'. (A song in Lionel Bart's Maggie May starts, "Dey Don't Do Dat Today Den, Do Dey?").

Both the voiced 'th' and the voiceless one (as in 'thin') are quite rare phonemes, existing in only 4% of the world's languages. Jamaican Patois has no voiced dental fricative, and in London it’s not unusual to hear younger Jamaican-Brits substituting 'v' for the voiced th ('Get anovver one') and ‘f’ for the voiceless one ('I’m in your barfroom.’) This is also what many Cockneys do. It's known as ‘Th-fronting’ and it appears in several dialects of English.

In London at least, ‘v’ is increasingly being used to replace ‘th’ at the start of a word - ‘these’ → ‘vese’ etc. - a trend first noticed in a survey in the early 90s.

So yes, your 'dat' would be noticed and maybe ascribed to a dialect. If you find it hard to say "that", you might try "vat".

No. In a few accents and/or dialects, you will hear a soft "th" sound pronounced as a "d" sound, but that has nothing to do with how fast the speech is but only to do with the particularities of that accent and/or dialect.

You say you notice in "fast speech 'that' always sounds as 'dat,'" but that's not anything that actually happens, not anything native English speakers actually do, so it only has to do with your ear not yet being attune to properly distinguishing that particular sound when people are talking fast.

You apparently are still unable to perceive the nuance between a soft "th" sound and a "d" sound when the soft "th" sound is said quickly, understandable since the two sounds are so close, especially if your native language is Spanish since the Spanish "d" sound is halfway between an English "d" sound and an English soft "th," in Spanish, the "d" sound being formed by placing the tip of your tongue against the back of your teeth or even between your teeth, like is done when English speakers pronounced a soft "th," and pressing the area of your tongue behind the tip against your soft pallet. In English, the tip of the tongue isn't placed between or against the back of teeth but is placed against the soft pallet to make the "d" sound, but the tip of the tongue is placed between or against the back of the teeth to make the soft "th" sound." As a result, what a Spanish speaker hears as a "d" sound in their native Spanish sounds very much like English's soft "th" sound and very much like English's "d," the Spanish "d" sound being a sort of hybrid that is halfway between these two English sounds and so often makes it very difficult for native Spanish speakers to distinguish an English "d" sound from an English "th" sound, especially when spoken quickly, not unlike how Mexican Spanish speakers have a difficult time distinguishing between "b" and "v" when spoken by other Spanish speakers as Mexicans pronounce both "b" and "v" the same and in a manner that is halfway between how other Spanish speakers pronounce "b" and "v." So, because of how "d" is pronounced in Spanish, because of how a Spanish speaker has heard the "d" sound said all their life and the sound they associate with it, it makes it very difficult for native Spanish speakers, especially beginners, to distinguish between English's soft "th" sound and "d" sound.

At any rate, as your ear for English improves over time, so will your ability to properly distinguish the sounds that people actually make and the nuance between a quickly said "th" sound and a quickly said "d" sound will become audible to you, become increasingly perceptible to you.

Now, there are specific words where the "th" sound is pronounced irregularly by some native English speakers, an example being British people pronouncing the name "Anthony" as "Antony," but I can't think of any specific word that involves an irregular soft "th" that as a matter of course begets a "d" sound, instead, because speakers who employ a "d" sound for a soft "th" do it across the board with all such words as it is part of their accent or dialect.