I’m wondering if it’s a bad idea to use weird characters in my code. I recently tried using them to create little dots to indicate which slide you’re on and to change slides easily:

There are tons of these types of characters, and it seems like they could be used in place of icons/images in many cases, they are style-able and scale-able, and screen readers would be able to make sense of them.
But, I don’t see anyone doing this, and I’ve got a feeling this is a bad idea, I just can’t decide why. I guess it seems too easy to be true. Could someone tell me why this is or isn’t okay? Here are some more examples of the characters i’m talking about:
↖ ↗ ↙ ↘ ㊣ ◎ ○ ● ⊕ ⊙ ○ △ ▲ ☆ ★ ◇ ◆ ■ □ ▽ ▼ § ¥ 〒 ¢ £ ※ ♀ ♂ &⁂ ℡ ↂ░ ▣ ▤ ▥ ▦ ▧ ✐✌✍✡✓✔✕✖ ♂ ♀ ♥ ♡ ☜ ☞ ☎ ☏ ⊙ ◎ ☺ ☻ ► ◄ ▧ ▨ ♨ ◐ ◑ ↔ ↕ ♥ ♡ ▪ ▫ ☼ ♦ ▀ ▄ █ ▌ ▐ ░ ▒ ▬ ♦ ◊
PS: I would also welcome general information about these characters, what they’re called and stuff (ASCII, Unicode)?
There are three things to deal with:
1. As characters in a sentence/text:
The problem is that some fonts simply do not have them. However since CSS can control font use you probably will not run into this problem. As long as you use a web safe font, and know that that character is available in that font, you should probably be okay.
You can also use an embedded font, though be sure to fall back on a web safe font that contains the character you need as many browser will not support embedded fonts.
However sometimes certain devices will not have multiple fonts to choose from. If that font does not support your character you will run into problems. However depending on what your site does and the audience you are targeting this may not be a problem for you. Not to mention that devices like that are very old, and uncommon.
All in all it was probably not a good idea a handful of years ago, but now you are not likely to have problems as long as you cover all your bases.
It is important however to point out that you should never hard code those characters, instead use HTML entities. Just inserting those characters into your code can lead to unpredictable results. I recently copied some text from Word directly into my code, Word used smart quotes (quote marks that curve inwards properly). They showed up fine in Notepad++, but when I viewed the page I did not get quotes, I got some weird symbol.
I could have either replaced them with normal quotes
"or with HTML entities to keep the style“and”(“and”).Any Unicode character can be inserted this way (even those without special names).
Wikipedia has a good reference:
http://en.wikipedia.org/wiki/List_of_XML_and_HTML_character_entity_references
2. As UI elements:
While it may be safe to use them in many cases, it is still better to use HTML elements where possible. You could simply style some div elements to be round and filled/not filled for your example.
As far as design goes they are really limiting, finding one that fits with the style of your page can be a hassle, and may mean that you will definitely need to embed a font, which is still only supported by the latest browsers.
Plus many devices do not support heavy font manipulation, and will often display them poorly. It works in the flow of your text, but as a vital part of the UI there can be major problems. Any possible issue one of those characters can bring will be multiplied by the fact that it is part of your UI.
From an artistic stand point they simply limit your abilities too much.
3. What are you doing?
Finaly you need to consider this:
Those characters are text, they are for telling someone something. So ask the question: “What am I doing?” and then use what was designed for that task. If you are telling use them, if you are showing use Image, or CSS.