Need help on this English question.

First up all this is not a home work question. I have just some doubts about the correct usage of the words.

The below sentences are about to know the correct usage of adjectives,adverbs, gerunds and verbs.

1. Go slow or Go slowly-
Always ride you bike slow/slowly when you are in the traffic.

Late or Lately-
2. I failed in the exam since I got the admission(very)late/lately.

Both(adjective and adverb) can work here? How?

3.Is there a way to erase an existing/existed background displayed on the screen?

Would it be an alternative using a verb existed, instead existing?

4. Open the next webpage that lets your photo open.

vs

4. Open the next webpage which lets your photo open.

Can the word that also be used in same way like which?

Use of had to and would when talking about the past? Both have same meaning?

5. We all knew I had to get rid of him vs We all knew I would get rid of him.

Use of to + ing:

6. We’re on track to becoming(become) a developed nation.
or
6. I look forward to meeting(meet) you.

What is the exact difference when we use verb with ing after infinitive?

Your answers are really helpful!

Thanks and wish you a very happy new year!