You want to apply as a visitor to the Republic of Ireland. You claimed asylum in the UK because your girlfriend was pregnant; your claim failed and you were removed to India.
but I don't have those refusal letters for my appeals or even my
asylum refusal letter . Would that be ok ?
Yes, it's OK. But understand that they will ask the UK to send your paperwork, and the UK will dig it out of their archives. This could take up to two years after which your premise for visiting the ROI will most likely have expired.
Or what are the chance of getting refused becoz of no refusal letters.
They will not refuse based upon lack of documentation from you. As explained above, they will locate the paperwork first, and then refuse on the grounds that you are not a bona fide visitor.
Or other wise is it ok to have claimed asylum and expect to further
travel or that's it I can't travel Anywhere anymore ?
When somebody claims asylum it is tantamount to saying they fear persecution in their homeland and they accept the likelihood of never returning. This is the polar opposite of what a visitor is all about. To complicate things, you took the UK government to court after you knew you had a specious claim (this was a judgement error that they will not forgive because it needlessly wastes public funds).
Does this application of my asylum will be in immigration my
electronic record? Though I have a new passport .
Your passport has nothing to do with it (and why does it matter if your stated intent is to be honest?). You may (or may not) remember when you filed your asylum claim that they took your biometrics. When you apply to the ROI, your biometrics will be collected again and matched up. They will then assume that you want to enter the ROI and then use the soft border to the UK and then go underground. It's a fair assumption on their part and the onus is on you to rebut it.
Will the Irish embassy be able To view it when giving away the visa ?
Your stated intent of being honest makes it irrelevant what they can and cannot see. But as mentioned, the ROI will ask the UK to provide copies, and you can expect it to take a hellishly long time.
Since Irish visa application asks whether I was refused any visa
anywhere !! So I will have to write this asylum refusal wouldn't I ?
Or should I not ?
You can select either option, write it down or not, declare it or not. Biometrics made this sort of thing a lot easier for both options.
I don't have a deportation symbol in my passport though. So what do
They do not deport failed asylum seekers, instead they issue administrative removal orders. You wrote that your ban is nearing expiry. The absence of a ban doesn't mean you will have a successful application however. Your best option at this point is to visit the USA and other places with closely managed visa regimes, like Schengen.
I know you have other questions on the site where you use comments introduce new material and to debate the merits of your case. It's not how the site is modelled. We are not the Upper Tribunal and we're not the Garda Síochána; questions about the merits of a given case are heavily opinion-based. See
https://travel.stackexchange.com/help/on-topic for more info.