(idiomatic) to be sharp; to be keen; to be on the ball; to have a mind like a steel trap; to be clever

直ぐに: immediately

直ぐ: immediately, right away, soon / 直ぐ: quick to / 直ぐ: right next to, right close by

straight, not bent

(obsolete) 過ぐ: classical form of modern 過ぎる (sugiru, “to go through, to go past; to exceed”)

(obsolete) 挿ぐ: classical form of modern 挿げる (sugeru, “to connect or affix one thing to another, especially by passing something through something else”)

be cut off, break, snap, burst, collapse / be cut, be injured, become fissured / break off, be interrupted / cut or sever connection with / run out, expire, terminate

cannot completely... (because it's excessive)

Same as above.

be able to do, be able to finish