Should automatic tests interact with the outer world?

That rather depends on what type of automatic tests you're talking about. If they're unit tests, then no - they should be self contained following FIRST principles.

As for whether you write other types of tests e.g. integration tests - well, think very carefully about this. They tend to be brittle, hard to write, difficult to verify, and are long running. There is a school of thought as to whether these are worth writing at all. I think this is a bit of an extreme view, but the general premise is worth considering - see if the testing can be done in a more cost efficient way.

However, they can be cheaper than deploying each time and smoke testing manually. Like many things, it is a trade off. If you can prove you have a working system be having only a few of these integration tests, it may be worthwhile. If you're still getting lots of failures despite these tests, then you need to investigate other avenues.

Should automatic tests interact with the outer world?

That rather depends on what type of automatic tests you're talking about. If they're unit tests, then no - they should be self contained following FIRST principles.

As for whether you write other types of tests e.g. integration tests - well, think very carefully about this. They tend to be brittle, hard to write, difficult to verify, and are long running. There is a school of thought as to whether these are worth writing at all. I think this is a bit of an extreme view, but the general premise is worth considering - see if the testing can be done in a more cost efficient way.

However, they can be cheaper than deploying each time and smoke testing manually. Like many things, it is a trade off. If you can prove you have a working system by writing only a few of these integration tests, it may be worthwhile. If you're still getting lots of failures despite these tests, then you need to investigate other avenues.