You have to ask the Halifax - I can think of no reason why they should suddenly cancel your credit card when you are paying responsibly. A couple of our card issuers have reduced our credit levels (we were not using all of it) and we could see the sense in that, on a risk exposure basis. As you have an existing card, IMO then they must tell you why they are suddently cancelling it - it sounds to me that if they are not cancelling it, just putting a block on you using it any more. The interest rate is very high, more that we pay on our Halifax card. Have you undertaken a credit check with somewhere like Experian? That might throw up if there is incorrect info on your credit record.
Lots of the card providers are now being far more responsible for trying to help customers reduce their balances and not add more debt.
If you apply somewhere else, I couldn't say yet whether the declined further loan from Halifax will show up as it may just have been dealt with internally - but yes it might. I suspect that Halifax have looked at your income, seen you paying regularly, but decided that enough is enough. I am sure that you are not alone - these sort of worries apply to all of us carrying balances on our credit cards. Paying the minimum will take you more than 20 years to pay off (I saw that on the TV the other day) so I can see why they don't want you to have any more credit.