They are not entitled to legal fees unless a contract between you provides for it or a statute provides for it (and there isn't one that would apply to this). As for their other claims, I doubt they can recover the cost of the land survey and they have the burden of proof on their claims. You should check with your home owners insurance company to see if you are entitled to have them defend and indemnify you for their counterclaim.
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No, typically people don't get into trouble for "lying" in court, unless it's outright perjury, which is knowingly saying something someone else can prove to be false and the person saying it can be shown to have known it to be false, which is a high standard.
Usually what happens is that in civil litigation, it's often like a backseat sqabble between litigantsof the "yes you did" "no I didn't" or "he said" "she said" variety, with a trial for a judge or jury to determine whose version of the facts and law is most credible, but there's no downside for "lying" unless you can prove it was a knowing lie, which is very difficult if not impossible (like that person said something different to some other witness, so you can show they were uttering falsehoods under oath).
Possibly what's more problematic in your "tit for tat" litigation with your neighbor is that usually money damages aren't claimed for trespasses of this sort by either side. Usually, what you're seeking is a court declaration of where the boundary line is and for one party to stop blocking access to the other (an injunction), if the court determines that trespass was involved. Where did you come up with this $500 number and what is it based on, if anything? Ditto with the survey stakes and lawyer's fees counterclaim.
Only in very limited actions can you get lawyers fees from the other side, and survey stakes usually only show where the underground pipes or rods used as survey corners are located for the purposes of imminent construction (then it can be a crime to pull up the boundary markers to harass an other person).
While you say you "consulted" with a lawyer, it doesn't sound to me like a lawyer is actually representing you in this action or signed his name to the papers as required to indicate this is not a frivolous claim (22 NYCRR Part 130). I'm sorry, but while there may be a valid boundary line or trespass suit somewhere in all of this, both the way you are proceeding for money damages and the "tit for tat" counterclaims of your neighbor looks way off base from anything that could be a legal claim on which a court might validly grant relief.
If you are interested in proceeding pro se, you may be interested in looking at various remedies in the NY Real Property Law and Real Property Actions and Proceedings Law, which you can find in your public or law library or online here http://bit.ly/10VlBxB . Or you might actually retain a lawyer to represent you in litigation, rather than just "consult" with one.
Lastly, while the most likely remedy for this litigation is to have it completely dismissed against both sides unless you can prove the monetary damages you are seeking, and it's unlikely a court will order attorneys fees be paid by one side to the other by pro se litigants, if your opponent had to hire an attorney to defend against your claim, I'd say you actually are in greater danger of paying the other sides fees, since bringing an action determined to have no legal basis simply to harass or annoy another party **is** one of the limited grounds for the winner's attorney fees to be paid by the losing side.
I would consult with this attorney again as to these developments and either retain him or another attorney to manage this litigation if you feel you have a valid claim after getting more advice on possible remedies.
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