Can the adverb 'quickly' be used here? He couldn't do anything, as his friend had quickly paid the bill.

2) He understood the basis of capitalism and how it is suffocating the workers. (can we use 'and how' to connect it or is there a better way?)

3) In the dictionary, I saw the idiom keep in mind/bear in mind. Sometimes used with 'that' and sometimes not. Which is right? Bear in mind (that) she is crazy.

4) Is 'which' used correctly here? It was up to them to do the talking, which they did with ease.
His was a beautiful yet sad smile, which the father supposed was quite remarkable.

5) Are 'if' sentences always conditional? Are there exceptions? If he has so much wealth now, it's only because he worked hard in his youth. (here, if isn't conditional, right?). If their agenda was beneficial, maybe their motives were also pure.

If isn't always conditional. This is how RHUD defines and explains it:
if, conj.
1. in case that; granting or supposing that; on condition that: Sing if you want to. Stay indoors if it rains. I'll go if you do.
2. even though: an enthusiastic if small audience.
3. whether: He asked if I knew Spanish.
4. (used to introduce an exclamatory phrase): If only Dad could see me now!
5. when or whenever: If it was raining, we had to play inside.

IF, PROVIDED, PROVIDING imply a condition on which something depends. IF is general. It may be used to indicate suppositions or hypothetical conditions (often involving doubt or uncertainty): If you like, we can go straight home. If I had known, I wouldn't have gone. IF may mean even though: If I am wrong, you are not right. It may mean whenever: If I do not understand, I ask questions. PROVIDED always indicates some stipulation: I will subscribe ten dollars provided (on the condition) that you do, too. Provided he goes, we can go along. PROVIDING means the same as PROVIDED, that is, just in case some certain thing should happen: We will buy the house, providing (provided) we can get a mortgage.