Do you think having no exit record from the UK on my passport will risk my visa application for re entering?
No. The UK does not place exit records in passports. Nobody has such a record, and not having such a record couldn't possibly have a negative impact on your visa application.
As noted in the other answer, the UK does record certain departures electronically, but that system began in 2015, and from your question it sounds like you left before it went into operation.
If you left after it went into operation, it is possible that they overlooked your departure (because some classes of departure are not recorded), but very unlikely.
If they do find that you overstayed when in fact you did not, you can subsequently present evidence of your presence outside the common travel area to challenge that finding. I don't think you should do that in your initial application, however: every extraneous piece of evidence you include in your application presents additional risk of giving the visa officer the impression that you are trying to convince them of something that isn't true. In other words, presenting an argument against a position that the visa officer wouldn't otherwise take might in fact prompt the visa officer to take that position.
It could work against you.
Instead, it's best to present the minimum amount of evidence necessary both to satisfy the requirements in the application instructions and to establish that you meet the requirements of the visa.
In discussing immigration interviews, people often say "don't say anything unless asked, and always tell the truth." This is true of visa applications, too. Respond to the questions in the application truthfully and completely, and do not give extra information to address additional matters unless you know that the visa officer will in fact be concerned about those matters.
I am extremely skeptical about my chance of obtaining the visa because of the dramatic immigration history. Please kindly share your opinions about these matters.
Asylum is of course a touchy subject for many, and in the UK this has manifested itself as a "hostile environment" toward immigrants. Your history as an asylee in France should not have a large bearing on your application, however, because you are now a French citizen, and the chance of your seeking asylum in the UK is virtually nonexistent (as would be your chances of success as a French citizen).
Do you think it is ideal to retrace past history, even though I was not involved in anyway in crime, and obeyed responsibly the rules as a student during my stay in the UK.
Whether to "retrace" the history depends on the questions in the application, as noted above. If they ask about your past stays in the UK, which I am fairly certain they do ask, then you must give the dates and the circumstances of your studies in the UK. They will probably ask about your current and past citizenships. You should answer truthfully. If they ask you about how you became a French citizen, tell them. If they don't, don't.
Once again, answer every question completely and truthfully, but do not volunteer extra information.
Above all, do not lie. The UK takes a very dim view indeed of misrepresentation. If you lie, you run the risk of being banned for 10 years, and after the 10-year ban expires you will find it virtually impossible to get a visa because your credibility will continue to be severely damaged.