Decompress the overlays, use hex editor's search function to find the left side and replace with what's on the right.

Here are the overlays for those who want them:

Stop timer:
overlay 0

Always play as Luigi:
overlay 9

Don't switch screen even if entrance set to (imperfect)
overlay 54 (first one)
overlay 10 (second one)

Disable challenge mode:
overlay 8 (both)

Change the amount of fireballs Bowser Jr. can withstand from 8 to 18 (NSMBW style)
overlay 28

Make groundpounds deal the same amount of damage as normal jumps to Bowser Jr. (partial)
overlay 28

Change what world Mario will be sent to upon clearing a castle:
overlay 8


EDIT:
This is an update about the world progression:

Overlay 8

Function	Length	US offsets	EU offsets
Route upon clearing W1 Castle	1 byte	0×1FF8	0×1F0C
Route upon clearing W2 Castle	1 byte	0×2004	0×1F18
Alternate route upon clearing W2 Castle	1 byte	0×2008	0×1F1C
Route upon clearing W3 Castle	1 byte	0×2010	0×1F24
Route upon clearing W4 Castle	1 byte	0×2018	0×1F2C
Route upon clearing W5 Castle	1 byte	0×2024	0×1F38
Alternate route upon clearing W5 Castle	1 byte	0×2028	0×1F3C
Route upon clearing W6 Castle	1 byte	0×2030	0×1F44
Route upon clearing W7 Castle	1 byte	0×2038	0×1F4C
Power-up required for W2 Castle alternate route	1 byte	0×2000	0×1F14
Power-up required for W5 Castle alternate route	1 byte	0×2020	0×1F34

World	Value	Note
1	00
2	01
3	02
4	03
5	04
6	05
7	06
8	07

Power-up	Value	Note
Normal Mario	00
Super Mario	01
Fire Mario	02
Mega Mario	03	Useless
Mini Mario	04
Shell Mario	05

Very nice, I added the overlays to the OP.


I wonder whether you'd be sent to the appropriate world if you set the powerup byte to megashroom and touched the key with the mega, or if it only cares which powerup you end the level with.

Nope. Just tested and the game isn't smart enough to recognize that you touched the key (or the flagpole for that matter) while being Mega, since Mario automatically reverts back to Super.

What the game checks is Mario's current powerup on the map.
Forcing the powerup on 3 on the map does make the game consider the alternate route instead, but since you can't have Mega-Mario on the map outside of cheats, value 3 is essentially wasted.

I present you the damage table for the Bosses.
It was skawo's idea, and already did begin the work with Bowser Jr.
Also the table have some empty cells because trying to use rowspan on more than 1 row break the table on this site.

Boss	Overlay	HP	Incrementation	Data type	Weapon	Default damage	US offsets	EU offsets	Default data
(Final) Bowser Jr.	28	144	+	Byte	Fireball	16	0×4334	0×4160	10
					Mini-Pound Stomp	48	0×4520	0×4338	30
					Ground-Pound	72	0×4198 0×41E0	0×3FCC 0×4014	48
(Dry) Bowser Final Bowser Blue Shell counter	13	9 30 3	-	Byte	Fireball	1	0×8370	0×7F94	01
					Blue Shell ((Dry) Bowser)	3	0×8204	0×7E20	03
					Blue Shell (Final Bowser)	10	0×81F8	0×7E10	0A
					Blue Shell counter	1	0×8218	0×7E34	01
Mummi-Pokey	16	150	+	Byte	Fireball	10	0×3038	0×2D98	0A
					Mini-Pound Stomp	50	0×3710	0×33C8	32
					Blue Shell	50	0×2F54	0×2CC0	32
					Ground-Pound	75	0×32A0	0×2FBC	4B
Cheep Skipper	18	90	+	Byte	Fireball	10	0×23A8	0×2224	0A
					Mini-Pound Stomp	30	0×2644	0×24A8	1E
					Blue Shell	30	0×22B8	0×213C	1E
					Ground-Pound	60	0×208C	0×1F20	3C
Mega Goomba	14	270	+	Byte	Fireball	10	0×262C	0×23BC	0A
					Mini-Pound	30	0×2CC8	0×2A04	1E
					Blue Shell	90	0×2374	0×2130	5A
					Ground-Pound	90	0×2E34	0×2B78	5A
Petey Piranha	15	90	+	Byte	Fireball	10	0×42E8	0×4130	0A
					Mini-Pound (Head)	30	0×390C	0×3750	1E
					Mini-Pound Stomp (Lower body)	30	0×452C	0×436C	1E
					Stomp Blue Shell (Head)	30	0×38D8	0×371C	1E
					Blue Shell (Lower body)	30	0×4008	0×3E34	1E
					Ground-Pound (Head)	60	0×38B8	0×36FC	3C
					Ground-Pound (Lower body)	60	0×3D54	0×3B8C	3C
Monty Tank	19	12288	+	Word	Fireball	1536	0×1490	0×132C	06 1C
					Mini-Pound	4096	0×157C	0×1404	01 1A
					Blue Shell	4096	0×F78	0×E8C	01 1A
					Ground-Pound	8192	0×1268	0×1148	02 1A
					Bob-Omb	2048	0×1020	0×F20	02 1B
Lakithunder	17	150	+	Byte	Fireball	10	0×2C64	0×2AF8	0A
					Mini-Pound Stomp	50	0×322C	0×3044	32
					Blue Shell	50	0×2B80	0×2A20	32
					Ground-Pound	75	0×2E6C	0×2CC8	4B

Note 1: Putting damage=00 for any given weapon (fireball / stomp / mini-pound / blue shell / ground-pound) makes the boss immune to that weapon.
Note 2: Mini-Pound often uses the same variable than Stomp.
Note 3: During Bowser's battles, if Blue Shell counter increment is set to 0, every time Mario or Luigi slams into Bowser with the Blue Shell the score will go up by 32768 points (or 0×8000 in hex).
Note 4: If Blue Shell counter is negative, the score can glitch!
Note 5: Against Petey Piranha, if the damage dealt by a weapon is less than Max HP/3, the remaining health after the hit is more than Max HP×2/3, and the score is supposed to increase, then the score will glitch!


Tips:
Here are each value for Monty Tank in hex:
_1536 => 0600
_2048 => 0800
_4096 => 1000
_8192 => 2000
_12288 => 3000

Note how the second byte is always 00. Wouldn't it be easier if we only had to read to left byte?

Here are the rewritten values, from there you can modify them more easily:
06 1C => already good
02 1B => 08 1C
01 1A => 10 1C
02 1A => 20 1C

That's a very nice list there, but there's one thing...

Mega Goomba... I don't think it can actually be hurt by Mini Pounds. I think you just bounce off... So, where did the "30 damage" come from?

It does actually, hitting Mega Goomba 9 times while mini will KO it.

So it just doesn't do any animations? Weird...

Well, it would be kind of awful programming if you couldn't beat Mega Goomba just because the player happens to be mini and you were forced to lose a life because of that hypothetical technicality

Annnnnd… I knew I was forgetting something yesterday, the Bosses' behavior against the Starman:

Bowser Jr.:

Phase through.



Bowser:

Retains collision against a sliding Blue Shell, phase through otherwise.
Fireballs phase through.



Mummi-Pokey:

Phase through.
Sandballs retain collision.



Cheep Skipper:

Retains collision from above while out of water.
Retains collision against a sliding Blue Shell.



Mega Goomba:

OHKO if hit on the lower body (no points awarded).
Retains normal behavior if hit on the head.



Petey Piranha:

OHKO if hit on the lower body (4000 points).
Retains normal behavior if hit on the head.



Monty Tank:

The tank in itself retains normal behavior.
Retains normal behavior, except that a Ground-Pound deals the same damage as a normal Stomp.



Lakithunder:

Phase through.
Spinies retain collision obviously.



Dry Bowser:

Retains collision against a sliding Blue Shell, phase through otherwise.
Fireballs phase through.
Thrown bones retain collision.



Giant Bowser:

Retains normal behavior.
Fireballs phase through.
Homing Fireballs phase through.

I know it would be, but that's what it seemed like

Might be a dumb question but how did you go about finding those values  skawo?
For example, how did you find that time was "5C2042E2002081E5"?
Was it through the debugged No$gba?

Yes.

Could you explain how you go about doing it? I've looked for a few tutorials on here and on google but no luck.

An alternative would be to go and grab the NSMB IDA database from  Dirbaio (who have my thanks BTW), this is how I figured and tested most of the data.

______________________

Tempo hacks:

Time to activate the Fast Tempo:

Overlay 0
Data type: Qword
Relevant data: 64 (100)
Full data: 64 00 50 E3 08 00 00 CA*
Offsets:
0×A0BC (US)
0×99E8 (EU)


Actual Fast Tempo value:

Arm9
Data type: Word
Default data: 66 01 (358)
Offsets:
0×57474 (US)
0×473DC (EU)


Secondary Fast Tempo value:

Overlay 0
Data type: Word
Default data: 66 01 (358)
Offsets:
0×A4EC (US)
0×9E18 (EU)


Fast Tempo check on Area load:

Overlay 54
Data type: Word
Default data: 66 01 (358)
Offsets:
0×C644 (US)
0×B954 (EU)

*If the Fast Tempo check is different from the Secondary Fast Tempo value, then the Hurry Up tune will play on level start if the timer is set on 100** or less, not unlike New Super Luigi U.
This is the only reason you want to consider changing the Fast Tempo check at all.
Be warned though that in doing so the Hurry Up tune will play every time the game needs to load an Area (merely changing the View is not enough thankfully).

To circumvent a bit that problem change 64 00 50 E3 08 00 00 CA on Overlay 0 into 64 00 50 E3 08 00 00 1A; changing that lone nybble from C to 1 is enough to change the conditional branching from BGT to BNE.
Now the Hurry Up tune will play only AND only if the timer is at 100 (not 99, 50, 101, 300, whatever… just 100).
It doesn't resolve the problem completely since if you happen to change Areas when the timer is at 100, the Hurry Up tune will play again, but at the very least the problem is scaled down by quite a margin.

**Assuming default settings.

An easy quick way to do simple things is to go into Desmume's RAM Search, look what values change when you do whatever you're trying to change then go into No$GBA and put a breakpoint on when that value is changed.

Added a "make it so that Star Coins appear on the bottom screen quicker" patch.

Sry but... which overlay?
_________________________
Super Mario Bros. - The New Worlds | Full release
Super Mario Castle 2 | Full release

My Youtube channel
Best friends forever

Appears to be 54.

EDIT: You can see the difference here: https://www.youtube.com/watch?v=APEPkXSx49w

I'm not sure if it's a good idea to change this value, because if Mario has the required Power-up for the secret worlds, he takes the path where only Mini Mario fits after you defeated the boss.

EDIT: I just tested something. I changed the HP damage of Mummy Pokey so that you have to ground pound/stomp 5 times. But unfortunately if you groundpound on him 1 time and then stomp 1 time (or in opposing order), he'll be defeated too.

there should be a ROM Patcher in NSMBe with these patches loaded from a database as pre-made templates.