Yes, dead wood should always be removed once it's discovered, as it might harbor pests or pathogens that could affect other parts of the tree (not a guaranteed risk, since many of them are secondary and  not a threat, but it is a consideration) and because, when it falls, you don't want the branch tearing off healthy bark as it shears away. Plus, of course, there is a safety aspect of heavy wood falling from a height is a concern if it might damage property or harm someone.

A pruning saw, like those widely available as foldable models (where the saw blade folds into the handle or locks closed), is the simplest tool which also gives you more control over where and how the pruning cuts are made. Handheld versions are easiest to use for branches within reach, and pole-mounted versions are needed for a branch too high to reach from the ground. If you have an electric chainsaw, that would work too, but unless the wood to cut is very thick (not the case here), it's probably not practical to use for branch removal and can be more of a personal safety hazard.

It's hard to verify precision from the photo, but your mark for where to place the cut looks about right. You can use the information and diagram and linked video on our Pruning Trees web page (in the "proper pruning cuts" section) for guidance about how far away from the branch collar to make the cut. Basically, it should be as close as you can get without injuring the collar itself, since that holds the tissue that will most efficiently and effectively seal-over the wound over time. Don't treat the cut stub with anything; let it seal on its own. The three-cut method helps to avoid uncontrolled branch falling where healthy bark may be torn when the branch being removed is large and heavy. That approach is mentioned on the page linked above.

Miri