Validity of three syllogisms with venn diagram

Question

me and my study group are struggling with a "how to proof syllogism conclusions" approach. We got three syllogisms which look like the following:

We know for a fact that syllogism #1 and #3 are false. Syllogism #2 is true. This is how we solved each syllogism with a Venn diagram.

Syllogism #1

We drawed the following Venn diagram out of this:

This is how we reasoned:

1. No A are B, so we rule out all the spaces where they overlap together.
2. Not all B are C, so we don't rule out any A because there is a chance it can be an A.
3. Conclusion is therefore false because "Some C are A" where a C can also be a B.

Syllogism #2

This is how we reasoned on this one:

1. No A are B, so we rule out all the spaces where they overlap together.
2. Some B are C, this means not all so we don't rule out any of the B and C spaces.
3. Not all C are A, this is true because a C can also be a B and not only an A

Syllogism #3

This is how we reasoned on this one:

1. All B are A, so we rule out the spaces where a B is a B and where it is a C.
2. Some C are A, we added a circle between those because there must at least be one as a counterexample.
3. Some A are B, this is false because some A's can also be C. Therefore this conlusion is false.

Then, our questions are:

1. Is this the right way of reasoning for this syllogisms?
2. Is the usage of the circle applied the right way?

Please enlighten us with your view on this.

Kind regards, Jack