I don't know the dragons claw willow but I'm sure if it's a willow you can prune it as much as you like and it will keep growing .

I think it's one of those contorted ones, probably very easy to spoil its shape by pruning badly. It sounds very close to the house though.

What do you think if I moved it. It seems a good healthy plant I could get it about 12 or 15 meters away and I have a conifer that should go in its rough place about 6 meters from my house.

Any tree planted close to a house is not a good idea. Tree roots grow laterally and will be roughly the height of the tree so as the tree grows so do the roots which will then spread to drains and foundations. They can cause considerable damage which will give you a lot of expense to repair.

As for planting a conifer, you don't say what type it is but even dwarf ones don't stay small.

It will eventually grow up to 12 metres tall and possibly up to 8 wide so yes it is much too near the house, and as you presumably bought it for its attractive growth habit pruning could spoil the shape. In a border it would matter less but removing a tree from the lawn makes a mess of the grass. (Unless your tree was grafted onto a smaller rootstock - in which case you should be able to see a sort of lump under the first branches) If you want a small tree its better to buy one that will not grow tall as constantly restricting its height will make it bush out sideways and affect the whole shape.
Leaving the lower branches on will not prevent it growing tall.

Re the conifer as A. says it depends on what it is and whether you are prepared to take it out when it gets tall - many tend to go brown and bare at the bottom and become unattractive as specimens.

Had you thought about using a medium sized flowering shrub instead? whatever you decide look up the eventual height and width before you buy. Some things can be kept smaller by regular pruning and some can't.