Hi Beth,
You know what? English prepositions are among the toughest aspects for us English learners too. It's just not enough for us to just know the basic meanings/senses of English prepositions; we end up having to learn them with the words that go before/with them.
For example, we have to swallow things like "be good AT sth" and wonder why we can't say "be good ABOUT sth". But that's how English works when the key word is "good" and that should be the way we English learners take it.
Therefore, I guess the same thing applies to you too. Instead of trying to find a cure-all for a preposition (and chances are, there's none), deal with the key words first. Find out what generally goes with the key words. Here I'll list below the key words in your sentences and talk a little bit about how they are used in a Chinese sentence.
bored/boring → 无聊
common construction: Sb 觉得 sth/sb 很无聊。
The word-for-word translation can be: Sb think/find sth/sb boring. But it's also the way to say "Sb is bored with sth".
Or, you can place the object at the beginning to stress it and say: Sth/sb sb 覺得很無聊。
Therefore, "I am bored with this" is either "我 觉得 這(个) 很无聊" or "這(个) 我 觉得 很无聊".
deal with → 处理;应付 (应付 has the negative sense of stalling/hasting)
Unlike how "deal" needs a preposition to go between it and its object in English, we don't need a preposition to go after 处理/应付 in Mandarin.
Therefore, "I can't deal with this" can be "我 没办法 处理/应付 这个" or "這(个) 我 没办法 处理/應付".
red with anger → This is a very idiomatic one, and I guess there isn't a convenient counterpart in Mandarin. To convey the same idea, we tend to say "气得脸都红了". The literal translation can be "so angry that the face turned red". With "气得脸都红了", you have to know the "adj 得 S V" construction.