This one is right:

There is 1 apple and 1 orange available

This is wrong:

There are 1 apple and 1 orange available.

I would personally say

There is an apple and an orange available.


This is now wrong

1 apple and 1 orange is available

This is right (almost):

1 apple and 1 orange are available.

But again, I would say

An apple and an orange are available.


These are both correct:

There is an apple and 2 oranges available.
There are 2 apples and an orange available

So it depends on the number of objects the is/are is referring to first.

2 apples and an orange are available.
An apples and 2 oranges are available.

When it is at the end, it refers to the total number of objects.