Having spoken to a sailor friend, I think that I may have the answer to my questions!
The first point is that in the third verse the engineer says "Can't get steam for the fire in the flue". Surely, steam comes from the fire and hence the line should read "Can't get steam FROM the fire in the flue".
That being the case, with the steamboat losing steam and therefore power, it will start to drift and, if it's on a river rather than at sea there is a danger that it might drift into the river bank. Hence, Captain Tobin would want to check the depth of the river and that would be done by swinging the lead into the river and testing the water depth from the length of rope.
What do you steamboat experts think?
All the best.
"... screwing cotton all of the day!"