Hiking Rocky Mountain National Park

Hi guys, hope this is okay to post here. I will be in Denver June 2nd through to June 7th and I was planning on hiking Rocky Mountains on the 4th as I have a free day. Note that I am European and have never left Europe, so I really have no clue what I am doing here… For this reason I have a some of questions:

1. I think I would really enjoy hiking Rocky Mountains, but I am not sure if that is the best thing to do with a free day? Is there something else you’d recommend I do instead or perhaps in conjunction with hiking? Not a big museums guys tbh

2. What are the fees to hike something like bear lake to emerald lake? this was the path ChatGPT recommended me

3. Is it okay to hike with some beat up Rebook C 85 vintage? I want to retire these shoes either way, just want to see if I would put my feet and ankles in much danger if I use them

4. I want to get family and friends some gifts - would you recommend getting gifts on the rocky mountains gift shop? If there even is one…

5. What would be the best way to get to a hiking spot from downtown Denver? Bus maybe? Is public transportation good/recommended in Denver?

6. Is Rocky mountains the most recommend place to hike while there for my situation, specially given that I haven’t planned anything yet and its a couple days from now?

I’m sorry if these questions are out of place. I’m just a bit lost and any help is greatly appreciated!