It seems to me that "dóigh" and "súil" share a common meaning. For example:

Ní dóigh liom go mbeidh sí ann.
Bhí sé níos fearr ná mar a bhí súil agam leis.

Here both words mean "expect".

Does súil also mean "hope" and "looking forward" while dóigh can also mean "believe/my opinion"? It seems súil is a positive word while dóigh is neutral. Am I right?

As I learned it, their meanings are quite different, with dóigh expressing probability and súil expressing hope or desire, but it's possible that in some dialects words like these have taken on meanings not found in other dialects. Note that dóigh is used with the copula is, while súil is used with the verb bí, and they also use different prepositions.

Is dóigh leis go mbeidh sí ansin.
He thinks [it probable] that she will be there.

Tá súil aige go mbeidh sí ansin.
He hopes that she will be there.

Ní dóigh liom go bhfáighidh sé an post.
I doubt [= think it improbable] that he will get the job.

Níl súil agam go bhfáighidh sé an post.
I don't hope that he will get the job.

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

Thanks!