Since 行きましょう is the polite version of 行こう, your question boils down to how to use the volitional form properly in general. It's used in several ways:
- Similarly to English "Let's ～", it can be used to invite or encourage someone to take an action.
- When used for one's own action, it can express the feeling of "I decided to do ～" or "Let me ～".
- When there's no listener, it is used to prompt one's own action. It's like "Okay let's ～", "I made up my mind and ～", "Oh I should ～" or "Now I'll ～" said to yourself.
手伝いましょう and 手伝います are both natural expressions, but the former sounds like the person just decided to do it, while the latter can also be used to state a known plan. In English, the former might be closer to "Let me help you" and the latter to "I will help you".
A said 10万円あげましょう, which can be taken as an invitation ("Let's give them 100,000 yen") if the conversation is about what to give to a third person. But since there's no previous context, A's statement only means "Let me give you 10,000 yen". It's also okay to say 10万円あげます here (it sounds like a known plan/fact, but the difference is small).
On the other hand, B's response can be either 旅行に行きます or 旅行に行こうと思います, but he should not say 旅行に行きましょう (or 行こう) as a response here. In my intuition, if he said it at this point, it would sound like an invitation to A ("Let's go on a trip together"), which is inappropriate. It's difficult for me to explain why, but perhaps this is because it's not yet a situation for concretely self-prompting or deciding on a specific action or plan. Since B has just been offered 100,000 yen, there's no time to hesitate or make a decision. If B is already sure about how to use it, he would say 旅行に行きます, and if he is telling a likely use hypothetically, 旅行に行くと思います would be appropriate. After pondering or discussing for 30 seconds about how to use 100,000 yen, saying 旅行に行きましょう/行こう would sound natural.