HEY BRO! I have posted my own execution tricks or techniques in the past, but I think it’s 1 of those topics that can be addressed and RE-addressed, so I’ll give you my own ideas to try to supplement the other advice in this thread! I’m NO expert, but I have a lotta experience and work on my execution ALL the time (tonight at the arcade, for instance).
I ALSO used to prefer d-pads for specials, supers, and super cancels, but once I found a joystick I liked, I figured out gradually how to be better with it (equally important was the button layout, of course!).
I am HORRID on Japanese sticks (at least, on the 2p side), even if they’re supposed to be easier to use. I’m MUCH better w/ U.S. sticks. Oftentimes, if I can’t get something to work consistently or barely at all, if it’s not a stick I use regularly, I figure it’s the stick that’s the prob, and not me (broken or just NOT my kind of stick). Unfortunately, I’m often to subjected to broken sticks cos of fragile parts they’re made of, and/or the repair guy isn’t around often enough to fix or replace the parts 
I HAAAAAAAAAAAATE console sticks, especially! I only used ONE custom stick one time for console that I liked.
Once you find a stick that feels RIGHT to you (practice can help but some sticks are just BETTER than others), you’ve done HALF the work (I’ve been hunting on and off for the perfect home stick for a long time!).
HERE are my tips for execution: esp. the dp motions and super-cancels. I ONLY use these for the 2p side, cos it’s harder for me to super-cancel on that side, for some reason. These tips are for both sides, though, if helpful.
Some combos involving dp motions are tricky, mostly cos characters w/ dp specials often have qcf supers and/or qcf specials. For instance, you want to cancel cr.mk into lp srk with Ken, but you get cr.mk X fireball instead (“X” is cancel into special; “XX” is super-cancel). Even if the combo is safe to do if you mess it up (that example is not always a safe combo, depending on your opponent’s punish/counter options and how close you are to your opponent), you would have done more damage by cancelling into the dp/srk instead of the fireball (not to mention get a knockdown against your opponent, in that case)–right? 
The fireball comes out cos the you held down on the stick and then tried to move the stick from the crouch/down position to the toward position on the stick, and you might get a fireball or a super instead. Going from down to towards without accidentally doing a qcf rotation is tough.
I cancel from a crouching attack into an srk move by holding straight down, pressing the button for the cr. attack, then let the stick go back to neutral for a SPLIT second, then do an srk as fast as possible. This ALWAYS works for me on the 2p side, but sometimes I get the wrong move on the 1p side, so it’s not perfect for me, but maybe that’s cos I don’t use Ken much, and cr.lk, srk (electric shock) for Necro is not that important to my gameplay (esp. since I seldom use SAIII with Necro :P).
I ALSO used to have more trouble canceling a standing attack into super, than with canceling from a crouching attack. I figure this is because double qcf supers involve passing through the 2 lower quadrants of the joystick (from down to left is 1 quadrant; down to right is a 2nd quadrant, for instance). If you’re canceling from a standing attack, you’re probably often resting the stick at the neutral point, so you’re not already in any of the quadrants the stick has to pass through in order to do a super, so it feels like you’re “far away” from the rotation you need to do. What I do to cancel from a standing attack is make sure I’m doing a COMPLETE rotation for the super by starting the first qcf rotation from back, so I do a half-circle forward instead. Sometimes you just have to make sure you’re touching all the required points on the stick to activate the super (8 points on a U.S. stick, for example). Some say it’s a matter of hitting the diagonal point of the rotation (down-forward point that is required for most supers, since most supers in SF3 are qcf supers).
Super-canceling from a cr. attack is much easier for me, cos instead of actually TRYING to do 2 quarter circles forward, I just hold straight down, press the button for the attack, then do an SRK/DP motion, and I get the super instead of a DP special move because I NEVER let the stick rest in neutral! Going from down to toward counts as 1 rotation, and then doing the dp is the 2nd rotation. If I don’t have meter for a super yet, I generally avoid doing this to make sure I don’t get a dp special instead (unless I can hit-confirm it), but if the opponent blocked the attack you wanted to cancel, you usually don’t want to attempt doing a special or super anyway, cos you might get punished, unless there’s a lot of “pushback” from the attack (I think it’s called), so they’re too far away to punish you, or you’re trying to chip them to death (Dudley’s SAIII, Akuma’s SAII, e.g.).
You play NECRO?!?! 
I have SPECIAL problems w/ him, so my advice might be esp. useful here! If you want to cancel his b+mp or b+mk into SAI/SAIII, that might seem trickier, cos you have to make sure you press back to get the right attack first (neutral mk doesn’t cancel, for instance), and THEN you’re suddenly trying to cancel a qcF super from the BACK position on the stick. I treat these moves like the neutral attacks that can be canceled, such as Makoto’s neutral st.hp XX SAII. I just try to do the half-circle for the first rotation of the super, then a qcf for the next rotation I would SWEAR that her super is a double hcf, even though it isn’t. Sometimes I just have to PRESS HARDER on the stick to make something work better (I’ve been told this shouldn’t damage the sticks, either).
Once in a while, I get the wrong attack because I tried to start the super or special rotation too early. A perfect example is trying to do st.hk, srk w/ Dudley. Hit-stun often solves this problem though! In that case, his st.hk has a lot of hit-stun, so I can wait a split second after connecting the st.hk before I start the rotation, so I don’t get his t+hk overhead attack instead, since that doesn’t cancel into a special or super (only links into super, and of course, only on crouchers, so I wouldn’t want that attack to come out, in that scenario). In this case, I just needed to slow down on my inputs.
I hope this helps, and that I didn’t ramble much! 
The fireball comes out cos the you held down on the stick and then tried to move the stick from the crouch/down position to the toward position on the stick, and you might get a fireball or a super instead. Going from down to towards without accidentally doing a qcf rotation is tough.