Should the original installer fix my sprinklers for free?

No.

No - the sprinklers should have been installed following a survey by the former owners - they should not have assumed that their property line was in a place certain. That being said - did your neighbor have a survey done prior to fence installation? It may be worth your while to pay the money for a new survey so that you know exactly where your property lines are. If the neighbor's fence is erroneously on your land, in Illinois, after a certain amount of time such a fence would legally alter the property line. If your neighbor's fence is on your property, I suggest that you show them the survey and ask them to move the fence to comply with the property line. If you are on the losing side of the survey, you can either water your neighbor's yard for free everytime you water yours, or have the sprinkler company come out and cap off the sprinklers that are on your neighbor's property so that you don't. Good luck -

Provided the neighbors fence is actually on his property and not yours. Time for a survey. Did the neighbors fence contractor have a survey done? If not they could be completely wrong and need to move their fence. Otherwise, it depends on what the irrigation system installer did, if they had a survey done and it was wrong, then they may be able to go back to the surveyor and get some relief. I'd not count on them offering to fix the issue if you find the system was installed incorrectly on the neighbors property.

No, have you looked at the survey to be sure your neighbor installed their fence on their property? The old survey may have been wrong.