If you mean the fact that a loan application has been made then the answer is that this will almost certainly show up on a credit check. if you mean the personal details in the application then no, they should not be available to the public as this is personal data and is covered by the data protection act.
Personal loan rates are subject to fluctuation. most lenders will quote "typical" rates which in theory must apply to a majority of borrowers but that doesnt mean you will qualify for them. a good credit record could mean a better rate whereas a poor one could mean a worse rate.
as lenders can choose what rate to charge there is nothing to stop you applying for say a 9% rate but then only being offered a 19% rate. this could be because you are a bad risk of default or the lender might fear that you would repay them early (meaning less profit) and so they will charge you a higher rate to put you off or to make the profit in a shorter period.
There is no one single answer to this question as all lenders will operate different criteria but a good rule is at least 1 year but some lenders insist on 3 years and that the CCJ must have been satisfied.
it would make sense to have a chat with an IFA who might be willing to give you some initial assistance so that you can plan a realistic schedule towards your first home.
adverse events like CCJs or missed payments will have a much larger bearing when the amount of deposit you are putting down is relatively small (nowadays you would need more than 10% to counteract the existance of negative bureau data)
all lenders have their own requirements for income which still impact on the employed!
common requirements for the self employed can be full accounts drawn up by a qualified individual; self assessment return (the printed version HMRC post out not a home printed document); some lenders will ask for personal banks statements only. Some lenders will want all the above for the last 2 or 3 years.
this is a part of the filtering process that an IFA will look through when researching mortgage options for you and another reason to take advantage of someone with up to date industry knowledge rather than a pundit on a tv show or newspaper website column!