Well, there's a lot of loose threads here.
Pancake day - never heard of caroling or begging door-to-door then.It seems to be a winter thing (in my mind), All souls, bonfire night, Christmas.
Your Ladybird rhyme seems to be a mixture of several that I remember from childhood (London late 40s)

Not last night but the night before
3 tom cats came knocking at my door
I opened the door to let them in
And they knocked me down with a rolling pin

Ladybird, ladybird fly away home
Your house is on fire and your children are gone
(All but one whose name is Ann
She hid under the frying pan)

We used to recite this one (usually only the first 2 lines) if we caught a ladybird - hold it on our hand, and gently blow to make it fly off.

Teedy bear teddy bear turn around
Teddy bear teddy bear touch the ground
Teddy bear, teddy bear climb the stairs
Teddy bear teddy bear say your prayers
Teddy bear teddy bear turn out the light
Teddy bear teddy bear say goodnight
"Goodnight" (That was spoken, the rest chanted)
Teddy bear was a skipping rhyme (with a long rope, 2 turning, the others taking turns to jump and do the actions), or as a finger play at the back of the class (naughty 10 yr olds).

No, thinking about it, this was the one we did as a finger play -
This is the church (2 hands, knuckles interlocked)
This is the steeple (index fingers up and touching)
Open the door and see all the people (turn hands palms up, fingers interlocked pointing up)
Here is the parson going up stairs (make staircase with knuckles of on hand, 2 fingers of other hand climb)
Here he is saying his prayers (hands together)

Azizi, just like drawing a human 'family tree' you can get fairly wide spread before you've finished. Everyone ends up related to everyone else! What fun.