Is the statement that the two angels who record good and bad deeds are not affected by the presence of a dog in the house because they are "men" considered mockery of religion and leading to disbelief, and does it necessitate ghusl and uttering the two Shahadas?

You know best your intention and purpose when you uttered what you uttered. If you intended to mock the angels and ridicule the Sharia, then you must repent from this grave sin, for mocking Allah, or His angels, or His messengers is apostasy, and your sincere repentance erases the effect of the sin. However, if you did not intend mockery and thought that this was an answer to his question, then there is nothing upon you.