Last time vs. the last time +3 more questions

Hello! Can you help me with this?

1. I don't clearly remember where I stopped translating the last time.
Is the use of the definite artcile acceptible in this sentence? From what I know the definite article is normally used with the words 'last' and 'next' when we are talking about the past and can be ommited. What about my sentence? Does the same rule apply to it?

2. to weather well
In the sentence "The wall's weathered badly" the meaning is that the wall is damaged because of the weather, right? I'd like to know if it's possible to say the same sentence the other way round: The wall has weathered well - (the meaning being that the wall still looks good despite the weather). ???

3. I'm having problems putting the right article in the following sentences. In principal, they're very much alike, so I've decided to write a few of them here, so that I'll be completely clear on the use of articles in sentences like these:

-However, what happened points to the obvious conlusion that it was 75-80% people's fault rather than  failure of equipment.
-There was a large government organization responsible for refinement of "green" planes.
-In the event of failure of one part, another serves as a substitute.
-(this sentence's probably too lenghty) Lack of time which is an inevitable consequence of increasing speeds and altitudes, complexity of aviation equipment which consists of hundreds of units, environment changing with every second, and significant psycological strain all have led to the situation when a pilot experiences much too more emotional strain, fatigue, and tiredness than 40-50 years ago.

4. the week after next
Is it correct to say 'the week after next' in the following context?
-I'm leaving the week after next.

Or do you normally say: I'm leaving in two weeks. ???

Hi,

1. I don't clearly remember where I stopped translating the last time.
Is the use of the definite artcile acceptible in this sentence? From what I know the definite article is normally used with the words 'last' and 'next' when we are talking about the past and can be ommited. What about my sentence? Does the same rule apply to it? You can include it or omit it. Including it just makes the reference seem more specific, like you remember the last time more vividly.

2. to weather well
In the sentence "The wall's weathered badly" the meaning is that the wall is damaged because of the weather, right? I'd like to know if it's possible to say the same sentence the other way round: The wall has weathered well - (the meaning being that the wall still looks good despite the weather). ??? Yes, you can.

3. I'm having problems putting the right article in the following sentences. In principle, they're very much alike, so I've decided to write a few of them here, so that I'll be completely clear on the use of articles in sentences like these:

-However, what happened points to the obvious conlusion that it was 75-80% people's fault rather than  failure of equipment. OK. But you could also say 'an equipment failure'. 

-(this sentence's probably too lenghtyYes, it is ) Lack of time which is an inevitable consequence of increasing speeds and altitudes, complexity of aviation equipment which consists of hundreds of units, environment changing with every second, and significant psycological strain all have led to the situation when a pilot experiences much too more emotional strain, fatigue, and tiredness than 40-50 years ago. The sentence needs some work, but the articles are basically OK

4. the week after next
Is it correct to say 'the week after next' in the following context?
-I'm leaving the week after next.

Or do you normally say: I'm leaving in two weeks. ???

In Canada, you can say both, but the meanings can be different.

Pretend today is Wed. 1/July.

I'm leaving the week after next. I'm leaving sometime in the week beginning Mon 13/July.

I'm leaving in two weeks. Unless I am speaking imprecisely (which people often do), this means that I'm leaving on Wed 15/July.

Best wishes, Clive