"weight" and "weigh"

Weighting is something we do to adjust the proportional contribution of something. We apply a weighting factor to modify something's relative importance. Eg. in the cardboard canoe building contest, the race score was weighted higher than the presentation score and team cheer score when calculating the final score.

Weighing implies a direct comparison of things, just like on a pan balance. After weighing the options, we decided to skip the cardboard canoe building contest this year and go visit some friends.

Weighting means to adjust something's importance.
Weighing means to assess something's importance.

In your first example for weight - "and to weight that cost against" - weight means to make a direct comparison to some other thing, as if placing it on a pan balance already containing the other thing. But it is a very rare usage. "and weigh that cost against" would be the normal way to say it.

In your first example for weigh - "not able to weigh down" - weigh down is a phrasal verb with a distinct meaning. It doesn't really fit your question.