I’m no expert, so hopefully someone else will be able to back me up. But I think it’s more helpful to try not to think about the english letter sounds of the sound you’re making. For example, with Ta, it sounds to me like you’re talking about the small burst of air that comes between the “t” sound, and the ‘a’ vowel (which is when the vocal chords start moving)
In english, if we say “ta”, you might notice a small burst of air between the ‘t’ and ‘a’ sound as well. This isn’t just chinese. You may be hearing this as a ‘h’ sound. I think this also applies to your other examples.
Don’t forget, chinese people aren’t thinking of their language in terms of how it relates to english. And Pinyin was designed by chinese people, for chinese people. For me i found it more helpful to mentally map the written pinyin to the particular sounds in my head, rather than trying to read pinyin as english.
You should definitely watch the “Say It Right” series on this site, it’s incredibly helpful, not just with tones but all aspects of pronunciation.