I have a (joke) template file for writing papers, which contains in it
Lemma (Main technical lemma)
Let $D$ be a domain in (INSERT SPACE HERE) such that the following properties hold:
- Technical condition 1
- Technical condition 2
Then $D$ is both opened and closed in (INSERT SPACE AGAIN).
What is the point of the above? At some point in your mathematical career you will come to the realisation that the proofs you personally are going to write are all based on the few small set of technical arguments. For what I do the main tool happens to be the Method of Continuity. By the time you have this realisation, it will also be completely obvious to you how to formulate a given proof to fit the template.
But how to you come to this realisation? My only suggestion is to read more papers/books/proofs and write more of them yourselfs. Just like a foreign language, the only way to get better and converting your intuitive ideas into formal arguments is through practice and immersion.
As an aside, from the examples you gave, it is not quite clear whether your difficulty is with implementing $\epsilon$ - $\delta$ s, or with setting up the proof by contradiction.
Now, besides the usual proofs in textbooks, a good place to read up on proof techniques is Proofs from the BOOK by Aigner and Ziegler. Try to really figure out the details of each proof so you can explain the idea of it, a few days later, without having the book open. Another good resource for problem solving techniques is the Tricki. Of course, reading up on answers (and providing them) on this website would also help.
A bit on the quantifier issue. First you need to mentally nest the various implications. For example, diagrammatically I think (the following is not formal logic notation) uniform continuity to be something like
$$ \forall \epsilon \to \left( \exists \delta \to ( \forall |x-y| < \delta \to |f(x) - f(y)| < \epsilon ) \right) $$
So to contradict it by example, you want something that satisfies the hypothesis $\forall \epsilon$ for some $\epsilon_0$ but not the conclusion $\exists \delta \ldots$. Which means that for $\epsilon_0$ there cannot exist a $\delta$ with the requisite property. Which means that for such $\epsilon_0$ and for each $\delta$ the requisite property must be false. So after step 1 we have
$$ \exists \epsilon_0 \forall \delta \to \mbox{ not } ( \forall |x-y| < \delta \to |f(x) - f(y)| < \epsilon_0 ) $$
If the statement inside the parentheses were to be false, since it is a $\forall$ statement, you just need one example. So you can convert the negation to
$$ \exists \epsilon_0 \forall \delta \to \left( \exists |x-y| < \delta \mbox{ and } |f(x) - f(y)| > \epsilon_0 \right) $$
So there, we have converted a "negative statement" which we want to contradict, to a "positive statement" of a property we want our function to have.