1- choking on her tea
She was drinking tea and it went down the wrong way, so she started to choke.
2- let's run through your part
Let's go through/practise the role you'll be acting in the movie or play.
3- they all (drew) near the fire.
They gathered/came together near the fire.
It should be safeguard. It means to gaurd something against destruction or danger, to make it safe. Sofa fabric is safeguarded. If you spill something on the fabric it won't ruin the material.
re- means to do again, and united means to come together.
That isn't a word in the English language. Sorry.