UK visa refused due to sloppy documentation. How should I reapply?

I'm a Russian national reapplying for UK general (6 month) visa. Unfortunately I treated my first application with unacceptable sloppiness and only now come to realize seriousness of consequences. Having that said, I'm going to quote a piece of refusal letter:

• You have stated that it is your intention to visit UK for one week for tourism
• You have stated that you are employed as ... for ... with salary .... per month. Your application does not demonstrate that you are employed as stated
• You have submitted a one page bank letter from ... stating that there are available funds of ... $. This letter doesn't demonstrate the providence of the funds nor does it demonstrate that you are in receipt of any income from your claimed employment.
• I am not satisfied that you are genuinely seeking entry to the UK as a visitor for the period and purpose as as stated or that you intend to leave the UK on completion of your visit. Paragraph 41 (i) and (ii).
• I am therefore not satisfied that you have demonstrated that you will be able to maintain and accommodate yourself adequately in the UK or that you can meet the costs of your onward or return journey from the UK. I have therefore refused your application because I am not satisfied, on the balance of probabilities, that you meet all of the requirements of relevant Paragraph of the UK Immigration rules.

For the second application I'm collecting all documents possible, which are:

• Proof letter from my employer, letter from employer stating that I have paid vacation for dates in question
• Same bank statement + statement from another bank where I have more money
• Transaction history for the period of half of the year from both banks
• Aircraft tickets and hotel booking confirmation

I'm afraid of possible inconsistencies in first and second applications. In the first application I stated that I'm employed at BigFancyCorporation. And I do work there. But, BigFancyCorporation often hire new employees though some sidekick employment agency littleCo. (It's kind of popular situation here and I beg you to not go offtopic about acceptability of that).
I checked with both of them, and littleCo guys said that they can write a proof letter that I'm littleCo employee and mention there that I'm working for BigFancyCorporation. To be perfectly clear about my occupation - I've been to littleCo one time for one hour one year ago.

I would like to state in the second application same thing I did at first - that my employer is BigFancyCorporation. And attach proof letter with above mentioned content. I believe it's better than stating another employer in second application. It would just look like I've lied at the first time. I'm relying on your best judgements guys to figure out this problem.

Second question is simpler: How fresh must the documents be?

Especially, transaction histories. Because I would have to give them away for translation, it takes time. I couldn't find this information online.