Since

Yes, you can make that replacement as long as you are meaning it as a relation to time. If you were wanting to use it as a relation to reason, you would need to use because, not since. However, you are using improper grammar. You can not use do and since together because of incorrect subject-verb agreement. A more correct way of writing this replacement would be:
I haven't lik

Thanks for your helpful reply, I appreciate it. It's hard for me to understand why I cannot use do and since together. This sentence:

I haven't liked him since I saw him with my crush.

looks very odd to me. I've seen this for the first time. Can I substitute haven't liked with didn't like here? And what about anymore, is it also in

Do is present tense. Since is past tense. This is why it is improper subject verb agreement. You can not use anymore and since together because they are both adverbs modifying have not. You have to use them in separate clauses. You also can not replace haven't like with didn't because that places the action to where it is happening at a different time.

This is more complicated than I thought. In English, it seems to work the exact opposite way compared to my native language. Here's another example:

I use a japanese knife since I saw it on a cooking show

Is that correct? Or do I have to use past tense as in:

I have been using a japanese knife since I saw it on a cooking show

The second is correct. The subject and verb have to be the same tense. However, this is English, so there are always exceptions, such as gerunds, participles, and infinitives, but most native English speakers don't know how to use those, so I wouldn't worry about it.

Thank you, I think I understand it now.