Your first three sentences are correct (and are pretty much variations of the same concept), while the fourth one (with 'never') is incorrect.
Let's start by comparing these two:
This thingamajig is nothing like anything I have ever seen before.
This thingamajig is something I have never seen before.
Both of these carry with them the meaning of "it's new to me". The first sentence means basically, "I have seen many things in my life, and none of those things are like this thingamajig" - in other words, you're comparing the thingamajig to everything you've seen before in your life. The second sentence reads along the lines of "I have never seen something like this thingamajig".
Also, notice that both of these sentences only have one negative (nothing + ever, something + never).
Now, let's look at this incorrect sentence:
This thingamajig is nothing like anything I have never seen before.
First of all, notice that here you have two negatives ("nothing" + "never"), which makes for a convoluted sentence in and of itself (even without the extra fun that "anything" adds).
But also, what are you comparing the thingamajig to here? To the "anything" (ie. everything) you have "never seen before". So not only are you comparing it to things you've never seen, you're comparing it to everything you've never seen. This makes for a bizarre statement, and an illogical one as well (since the things you've never seen before also include the thingamajig and its twin sister).
Does that help?
Best Answer
We can say things like:
But we don't say "Do never do such a thing".unidiomatic
P.S. In contemporary English, the do never {verb} construction is either a formulaic literary holdover from the 17th century, a petrified expression, or a regionalism. One may still find it in religious texts, where the language tends to be conservative and resistant to change, and often imitates the phrasings of earlier exemplars.