Q1: No, I don't think you will always be able to find an assumption in an argument. There are too many possibilities. For example, I can say that I will get a burger from Inn-n-out today. There are many assumptions one can make (i.e. I have transportation to get there, Inn-n-out will still sell have burgers, I won't get hit by an asteroid, etc...).
Q2: Yes, for strengthen questions you would do the opposite and find information to support the argument.
Here is the approach I would take for strengthen/weaken questions:
ID conclusion and its supporting evidence
(1) ID the author's assumptions (if any). If present, prephrase something to prevent you from choosing a wrong answer choice.
AND/OR
(2) Add additional information to make the argument stronger/weaker.
Also, for strengthen/weaken questions the stronger claim is usually correct. You do not need to totally destroy an argument but merely strengthen/weaken it by 0.0000001%.
Consider the claim from Bobby, "eating lots of Halloween candy can cause cavities".
(a) Some people who eat lots of Halloween candy do not develop cavities. (this statement is still consistent with the claim that eating lots of candy can cause cavities) Eliminate - this claim does nothing to the argument.
(b) All people who eat lots of Halloween candy will eventually develop cavities. (This statement is right because it doesn't leave any gaping holes in the argument. When one eats candy, they will develop cavities)
HTH