Blyka's Door - SD Powered Up - Challenges Guide

Here are further details and words of advice for tackling the 18 Challenges. As a general tip, remember to save often for any long-term Challenge so that if you fail, you can reload your file and try again.
Oh, and this page contains SPOILERS for secrets in the game, so don't read if you want to find everything yourself.

01. Circuit Breaker - Clear the game on Normal Mode.
Simply beat the game.
02. Sizzling Circuits - Clear the game on Hard Mode.
Beat the game on Hard Mode, which is unlocked after clearing Normal Mode. The following is different from Normal Mode:
-No Fusion energy refill after stages or game over
-More enemies
-Enemies deal more damage (+2)
-Item drops are rarer
-Can't equip special items

Enemies can pile up fast if you just run through and it's easy to get overwhelmed, but if you proceed cautiously it's easier to pick them off one at a time. It also helps to learn enemy patterns and timing so you can clear them out more efficiently.
03. Net Battler - Clear the game on Fusion Mode.
Beat the game on Fusion Mode, which is also unlocked after clearing Normal Mode. Fusion Mode is identical to Normal Mode except Cross Fusions have unlimited energy.
04. SUPERDANNY9 - Accumulate 9 lives and 9 D-Tanks at once.
This one often comes naturally just by playing through once you're skilled enough. Otherwise, find a good stage to farm for them with the Exit item.
05. Completionist - Clear the game with all items and Fusions obtained.
Each of the Robot Master stages contains a special item, all of which need to be collected, and note the specification "Fusions". There are two hidden bosses in the game, which each give a new Fusion. One is in the main 8 stages, and the other is in the first Wily Castle.
06. Slot Out - Clear the 8 Robot Master stages without using a Fusion in one run.
This includes the "stages" of the Robot Masters, so basically, don't touch any Fusions until all 8 Robot Masters are defeated.
07. Traditionalist - Defeat the 8 Robot Masters with Buster-only in one run.
All bosses can be defeated Buster-only without taking damage, so once you learn their patterns this should be pretty easy. If you don't Buster-only a Robot Master in their stage you can make up for it during the rematches in the second Wily Castle.
08. Plasma Power - Defeat all possible bosses with Buster-only in one run.
"All possible" is only excluding the Boobeam Traps and Alien Wily. This includes the two hidden bosses, and you have to buster-only the Robot Masters both times you fight them. Again, all bosses can be defeated without taking damage once you learn their patterns, so this shouldn't be too difficult.
09. Eat Your Heart Out - Defeat a Robot Master with Buster-only without getting damaged.
Whoever's pattern you've learned best, go for it. Fire and Ice are probably my choices for easiest, but all of them are pretty easy once you learn their pattern.
10. Won't Miss This Time - Don't miss with the Buster and clear the game.
The trick here is to very carefully clear one stage with the Buster, and then use Fusions for the rest of the game. All up to choice, really, but if you've learned his pattern Quick is surprisingly easy to defeat without missing and his stage is pretty friendly for not using the Buster much, so he earns my recommendation.
Ricocheted shots don't count as a miss since you technically hit your target, and also keep in mind that if you see a shot has missed, you can save it by pausing quickly before it goes off screen.
Tip: While certainly possible on Normal Mode, I recommend doing this one on Fusion Mode for an easier time of it. ;)
11. A Fly On ze Wall - Reach a Robot Master's room without getting damaged.
This can be done anywhere, but if you want the easy route I recommend Quick Man's stage as it's dominated by instant-death obstacles, making for fewer enemies to hit you. If you get damaged you need to exit and re-enter the stage to try again (and yes, instant death of any sort counts as getting damaged).
12. Robo-Twit - Reach a Robot Master's room without shooting.
Again, since Quick Man's stage is dominated by instant-death obstacles, it works well with fewer enemies to hit you.
13. Anti-Gravity Gizmo - Clear the game without falling into any holes.
The shield item from Ice Man's stage should help a lot with this one. Just be careful.
14. No Dew Break - Clear the game without using any D-Tanks.
This is actually quite easy, but you may want to have a few extra lives and go for boss weaknesses to be safe. Check the Damage Chart for help on that.
15. Trying to Quit - Clear the game without picking up more than 8 energy pellets.
As you may have guessed the allowed 8 come from the boss rematches, so really, you can't pick up any energy pellets throughout the whole game (Fusion energies don't apply). Play well so that you don't need them, but most of all, be very careful not to jump into an enemy explosion or some such. Stay alert.
16. Meteor Speck - Clear the game without getting 'Game Over'.
Play well and farm lives if you need to.
17. Meteor Fragment - Clear the game without dying.
Play well, save often, equip items, exploit all the weaknesses, farm D-Tanks and don't be afraid to use them for the safest ride through.
18. Anti-Everything - Clear the game on WildBlue Mode.
WildBlue Mode is unlocked when all other 17 Challenges have been completed. The following is different from Normal Mode:
-Begin every stage and every new life with 1 energy
-No Fusion energy refill after stages or game over
-Spikes kill instantly, even when hurt/flashing
-Can only carry 1 D-Tank at a time
-Enemies deal more damage (+1)

By this point you should have pretty much mastered the game, and it really isn't as bad as it may sound. Keep in mind that energy is precious, and if you're fighting a boss and you're ready to die it's usually better to use a D-Tank than to die and try again, since you'll probably just be using it immediately to get full energy anyway.
If you manage to clear WildBlue Mode you're rewarded with several prizes, three of which I think are particularly fun and awesome, so good luck!