Whatever it is, it's a weeping one not an upright whatever it is!

The height of the tree is determined by the length of the stem on which the weeping part is grafted, so no it will not grow any taller. The graft has essentially stopped any upward growth.
Pruning is fairly easy, bit late now though, but in future, as soon as the catkins on any will around have finished, then cut back to about a foot from where all those branches begin. That way you get nice new growth which does not trail on the floor straight away.

I'd take the bottoms off now, if they're trailing on the ground - just remove them to an inch or so above ground and take more off after flowering next year.

Kilmarnock willow is weeping Cammomile but I think looking at the photos this one has been badly grafted. Agree with Bamboo cut of the bottoms of the branches now and give more of a prune next year after flowering.

Oh poor thing xx

NanW, they do get wider each year & the stems in the middle tend to die back. I used to trim mine around the bottom & my dog liked to lie inside it in the shade. I like the umbrella shape it makes.

Thank you so much for all of your helpful comments - I am so disappointed ! probably because of my being so envious. I am going to transplant it to another part of the garden and save up for a taller one.

Wait until late autumn to transplant Nanw. it won't take kindly to being moved now. Our Kilmarnock willow is over 20 years old and remains at around 5 foot in height. Sadly in the tree world you generally get what you pay for. The trade stands at Gardening Scotland sell very cheap Japanese Acers, they are cheap because they have been very badly grafted and they hope folk at the show won't notice. In general buy trees from specialist nurseries or GC's rather than from a supermarket where they are just another commodity.