Question

should however start a sentence

Answer 1

 Yes, however maybe start a sentence .You can put it anywhere in the sentence as you want, but However is a linking tool such as (But) and it can come at the end of the sentence or in the middle, but it is better if it comes in the middle as a linking tool, but it is placed anywhere, so do not worry about using However. 

Exampels of  however :

However, I don't think finding these solutions means an end to all our troubles

I absolutely must see him, however painful it may be for me.

On the river, however, the adventurers seemed to be perfectly safe.