The short answer is this: It's possible, but hard to pull off.
First, you would have to do some basic research on court filing procedures. It's not rocket science, but you can't just pop an envelope in the mail addressed to the judge.
Once you figure out how to file documents for the case, you need to determine what type of documents to file. E.g. a motion to dismiss might be appropriate if you think the claim is utterly bogus.
It can be "plain English," as Smattiso suggested, and indeed, judges always appreciate clarity. Legalese isn't required, but valid legal reasoning is. Contrary to popular belief, "legal reasoning" doesn't have to be arcane. It's pretty much just a matter of using everyday logic to apply legal principles to the facts of the case.
It's a lot like learning to program. There's no magic involved. It's just a matter of reading the relevant documentation, looking at examples, and using your innate reasoning abilities to put something together in accordance with the rules. And yet, as with programming, reasoning and writing about the law take practice to do well. You can hack together a simple CRUD app with zero programming experience and a lot of motivation. But you wouldn't want to bet your entire net worth on it. Likewise for legal writing. You can spend a few evenings of hard work learning the anatomy of a motion to dismiss and the basics of trademark law, and then you can write your motion. But I wouldn't do it if my personal finances were on the line.
In a situation such as the OP describes, there is an additional challenge. According to the OP, the plaintiff is using and perhaps abusing quirks of court procedure to his advantage. (I'm referring to the bit about default judgement and the alleged absence of certain documents.) That, too, could in principal be dealt with by a non-lawyer, but it would be quite difficult. You'd have to learn a great deal about the finer points of court procedure. This is especially difficult because it's not uniform across jurisdictions.
EDIT: You would also need to determine whether this would constitute unlicensed practice of law in your jurisdiction. Generally, you're allowed to represent yourself in court ("pro se.") But the rules about representing your business are a little more complicated. Sometimes it's OK, sometimes it's not.
So, yes, you can sometimes write something up and get a case dismissed without hiring a lawyer, provided you're willing to learn a great deal very quickly. But you'd be taking a tremendous risk. Probably not a good idea.
Just to be clear: I'm not at all recommending that the OP do this. I'm just answering the question posed in the parent comment. The OP really does need a lawyer, especially considering the plaintiff has allegedly shown a willingness to play procedural games.