It took us 7 weeks on the market to sell - 8 viewings (plus second viewings) and two acceptable offers, so we sold to the people who went under offer themselves first.
To be honest, I think that how long your house takes to sell with any agent is down to how appropriately you are priced. We priced realistically (£525K and sold for £500K). The valuations from local estate agents were "price it at £535 and hope someone mad might go above the extra stamp duty threshold, but realistically you'll prob need to sell for £500"). In contrast, we have friends who have a house we always thought was essentially the same value as ours. They put it on for £600 I think in November, no interest, the price has come down, they've changed agents twice and down and down and it is now on at £535. So it will presumably eventually sell for £500, having messed up the marketing for their house, and stigmatised it as 'the overpriced house'.
sinking Housenetwork did check out the chain (they phoned all the relevant state agents - they didn't check for mortgage offers for our chain, as everyone is mortgage free so there is no extra finance to raise - but I'm sure that is part of their usual remit). Like angnes, doing the viewings ourself meant we have an incredibly good relationship with our buyer so we largely did the negotiations ourselves, and DH and our buyer are in daily contact updating each other on our solictors' progress (the whole chain is very motivated to get an early exchange).
Murder I personally found it fine doing the viewings myself. It's good to go around and do a list of the features you want to highlight in each room - and it's the extra things that you as an owner can say that an estate agent wouldn't know that help sell a house. It's especially hard in the first room you show someone, as you haven't got to know the buyer yet and they aren't yet at the chatting-asking-questions stage, so get a 'script' at least for that room. So for my first room (living room) I pointed out - this room is lovely as the four windows get the light at different times of day - it's great sitting in the bay in early evening with a book as it's really sunny. This room is what sold the house to us in the first place etc". This usually led people on to ask how long we had lived there/why we were selling and from there the viewing flowed quite naturally.
I guess in your case murder you'll need to tread a fine line - just because you think the house needs a new bathroom/kitchen etc, other people might not think that, so you don't want to undersell and convince them otherwise! I would just focus on the positives - the light, the sizes of the rooms, the views, the work you have done, etc and leave them to make their own conclusions as to the rest.