I first think it's necessary to clear up something that has been confusing me from the start: we are actually talking about two different classes of words. The first is the original set of three: 大【おお】きい、小【ちい】さい、可笑【おか】しい. The rest if the words we're talking about are all different than these three.
The Special Three... And All The Rest
Why are these three special? To answer that, we have to take a look at the grammar of Japanese adjectival forms, what we commonly know as i-adjectives (形容詞【けいようし】) and na-adjectives (形容動詞【けいようどうし】). It's a bit more complicated than that in reality, but those are the biggest two sets and the most relevant to our discussion.
One thing that makes these three words special is that they are i-adjectives with a na-adjective form. That is also true of the rest of our examples, which I will come back to shortly. What is extra special about these three words, and no others1 is that they are incomplete na-adjectives that can only be used to modify a noun; they are like, broken.
All the other words that have been suggested, and indeed all those that could be suggested, have fully-fledged na-adjective forms that can be used in any situation a normal na-adjective can.
The Nitty Gritty & Some Examples
Conjugable words in Japanese (用言【ようげん】, essentially verbs and adjectives) have six different stem forms (活用形【かつようけい】). I'm going to talk about three of them, because they're pretty common with both types of adjective: continuative form (連用形【れんようけい】), terminal form (終止形【しゅうしけい】), and attributive form (連体形【れんたいけい】).
With normal adjectives, we can use all of these forms:
- ○ ゲームを安【やす】く買【か】えるお店【みせ】 (continuative form)
- ○ ゲームが安【やす】い (terminal form)
- ○ 安【やす】いゲーム (attributive form)
That's true of na-adjectives as well:
- ○ 幸【しあわ】せでいられる人【ひと】 (continuative form)
- ○ 静【しず】かに読【よ】む (continuative form)
- ○ その人【ひと】が幸【しあわ】せだ (terminal form)
- ○ 幸【しあわ】せな人 (attributive form)
There are a couple other ways to make some of these forms, but these are the prototypical examples. Now, here's the deal: the three words in the "special" class I discussed above (大【おお】きい、小【ちい】さい、可笑【おか】しい) can only be used in the attributive form. So we end up with something like this:
- × 可笑【おか】しで描【か】く (bad, continuative form)
- × 絵【え】が可笑【おか】しだ (bad, terminal form)
- ○ 可笑【おか】しな絵【え】 (good, attributive form)
This pattern is true for all three of these adjectives, and not for any other i-adjective. The difference is that with any of the other examples, all of the other stem forms are possible, for example:
- ○ 柔【やわ】らかに焼【や】く
- ○ パンが柔【やわ】らかだ
- ○ 柔【やわ】らかなパン
So, let me get back to the questions at hand.
Does the ～な usage really add a higher level of subjectivity?
Honestly, I don't know. Most of my research was on the usage and classification, not actually on the resulting meanings. It was suggested that the [n] sound in Japanese tends to add a softer feeling to words2, and that may influence the meaning here.
Are there any other i-adjectives which can be used as na-adjectives in this manner?
Yes and no... other than our Special Three, there are no i-adjectives that can be used as na-adjectives in only a limited capacity. There are, however, a number of other i-adjectives that also function as na-adjectives, for example, those suggested in a comment by @user1205935 (and I imagine there are any number more):
- 四角【しかく】い ⇒ 四角【しかく】な
- 真【ま】っ白【しろ】い ⇒ 真【ま】っ白【しろ】な
- 真【ま】っ黒【くろ】い ⇒ 真【ま】っ黒【くろ】な
Is there a system or set of criteria which can be used to identify adjectives which can be used in this way?
Since I've divided the words up into two groups, this warrants two answers. A research paper I found1 suggests that words which are both i-adjectives and fully-functional na-adjectives generally fall into one of several categories, including color, [edit this].
On the other hand, our Special Three seem to be unique, so there isn't much of a system, per se. And finally...
What makes the Special Three special?
That's a difficult one. I have yet to find a satisfactory answer; I suspect that no one knows, exactly. There are a number of theories (for example, that explained in @Kafka Fuura's answer) but there isn't anything approaching a real consensus. Unfortunately, this one is probably going to remain a mystery for now.
Now, there is actually a class of words which encompasses our Special Three: attributive words (連体詞【れんたいし】). There are actually quite a few of these words that can only be used in front of a noun, and they fall into several smaller subcategories: attributives ending in ～の (e.g. あの、この、その); attributives ending in ～る (e.g. いわゆる); and attributives ending in ～た (e.g. たいした).
Of course, our Special Three (the only attributives ending in ～な) function a bit differently than all of the others.
So for that reason, I don't much care for the classification of "attributive words" -- it combines a lot of words which work in very different ways together into one category, ignoring the underlying nuances. And the Special Three, which are even more special.
But you should probably know that category does exist. Technically.
1 Backhouse, A. E. (1984). Have all the adjectives gone?. Lingua, 62(3), 169-186.
2 Makino, S., & Tsutsui, M. (1989). A dictionary of basic Japanese grammar: Nihongo kihon bunpō jiten (Vol. 1). Japan Times (Tokyo, Japan).
Note: I can provide a copy of the Backhouse article if you're interested in reading it.