OK, here's a very short explanation. The normal mortgage process (whether a re-mortgage or a mortgage upon purchase) involves the following:
1. Lender send "mortgage pack" to solicitor. Existing lender sends Title Documents to solicitor on loan for inspection pending discharge of the mortgage. (or on a purchase, Vendors solicitor sends a copy of the title documents at contract stage - originals to be furnished on completion)
2. When solicitor gets title documents, he inspects them all, perhaps has a few questions for client, and then (usually) is satisfied as to title matters
3. Solicitor sends "Solicitors Undertaking" to lender. This in essence says: "I promise I will register a mortgage for you; I promise that the mortgage will rank as a first legal charge; I promise that the title against which it is registered will be a "good marketable title".
4. Solicitor registers new mortgage and registers discharge of old mortgage;
5. Solicitor lodges title documents with new lender, together with a certificate to state that it is a "good marketable title".
The Title Insurance process arises on a re-mortgage, but not on a purchase. It still (I presume) involves many of the same mechanical processes (i.e. registering the discharge of an old mortgage and registering a new mortgage). However, the title documents are not vetted; instead it is assumed that the solicitor who last acted in the purchase or mortgage of the property got things right. The risk that the solicitor didn't get things right (or, presumably, that problems have arisen since) is bought off with a form of insurance. This is what Title Insurance is in the context of a re-mortgage. (A different type of insurance - also called Title Insurance - can arise in the case of lost title documents and such like).
This procedure in theory lowers costs. I have my doubts, but time will tell: it is still a relatively new product.
Regarding your specific queries:
1. Stamp duty and Land Registry fees are the same either way - i.e. approx. €175 Land Registry and €1 per '000 stamp duty. Typical solicitors fee on a remortgage might be from €500 - €1,000 plus V.A.T.
2. In theory, not having an existing mortgage is not an issue; As long as the property has been bought or mortgaged once, then a solicitor will (presumably) have inspected the title documents and been happy with them, and it should be possible for the lender to proceed on this assumption, but insuring against the risk of title defect.
3. See long answer above.