There is a collection of paintings in my house.
There are many paintings in my house.
There is a lot of paintings in my house.
Are the above sentences correct grammatically?
''There are a lot of painting in my house''
Is it correct to use ''are '' in the above sentence?
As for the last one, yes it is very common either way.
I would use only 'are' with 'a lot of' + plural noun. For me, 'is' is unacceptable English.
The are a lot of (many) paintings in my house.
There isn't a lot of (much) space in the garage.