These are three rather different constructions, and different usage patterns (I hesitate to say "rules") apply. I will address them one at a time.
- After the house having burnt, he thought he was so lucky to survive.

The phrase "After the house having burnt" is not a natural form, and would not be used by a native or fluent speaker of English. What seems to be the intended meaning is best expressed, in my view, by:
1A) After the house burnt, he thought he was very lucky to have survived. 
The simple past allows one to discuss events at any time in the past, with no connection to any other event being required. Here the only connection is to the later event of his thought, and so the present perfect form does not work well. "Having burnt" is used to describe a condition continuing up to the present, or a condition the effects of which continue to the present. Thus it does not work well with "after" which places the related event subsequent to the burning.
The form "having burnt" could be used in a sentence such as:
- The house having burnt, it was no longer safe to occupy.

- Having burnt, the house was was no longer safe to occupy.

Also for no reason I can specify, "so" is not used as an emphasizer in the construction "he thought he was so lucky", although "he had been so lucky" is natural. but "very lucky" can be used in all places where "so lucky" can be, and many where it cannot.
The meaning of sentence 1 can also be expressed by:
- 1B) After the house had burnt, he thought he was very lucky to survive.

- 1C) After the house had burnt, he thought he had been very lucky to survive.

- 1D) After the house had burnt, he thought he was very lucky to have survived.

- After the house's having burnt, he thought he was so lucky to survive.
While not strictly wrong, the use of the genetive (possessive) in Sentence 2 is at least unusual, and I would avoid it.
The alternate suggested by MarcInManhattan:
2A) After the house's burning, he thought he was lucky to survive.
I like even less. I would prefer to invert this to
2B) After the burning of the house, he thought he was lucky to survive. 
Or one may use the genitive with a present participle, as:
2C) After the house's burning, he thought he was very lucky to survive. 
other variations are possible here.
-
- After having burnt the house, he felt guilt then and there.

- 3A) Having burnt the house, he felt guilt then and there.

- 3B) After having burnt the house, he felt guilt.

Sentence # is grammatical, but seems a little awkward to me. 3A emphasizes the guilt, 3B emphasizes that the guilt came after the burning. 3 tries to do both.
As another answer points out, this meaning can be expressed without using the perfect, but there is nothing wrong in doing so, that is a matter of style.
The irregular past form "burnt" was once the only past participle of "burn" The newer regular form "burned" is now more common, except in fixed phrases such as "burnt food' or "burnt timber". But there is nothing wrong with using "burnt", indeed I have a personal fondness for it and for related forms such as "lent" and "rent" (in the sense of tore, not leased).