Learn English – Verb for doing something unknowingly

The usual idiom is "I don't have to dig".

In speech it is possible to express "I can (not dig)" by intonation, but there is no customary way to manage that in writing. *

Edit in response to the edits to the question: in those cases where you really are talking about the ability not to, as opposed to the lack of obligation: no, you will need to use a perphrasis, probably "able not to", as in your example. There isn't an unambiguous way of using "can" for this meaning.

Slightly off topic: the scope of negation can vary a lot between different languages. English "must not" ("obliged not to") is not a translation of German "muss nicht" ("permitted not to").

Aside from the rather intricate rules for what to do with the understood subjects of the second verb in these complex sentences, mentioned by Lawler in his comment, there is also a 3 way choice of complement type. What follows "consider" in the illustration you gave is taken to be a sentence, schematically [Lucas consider [Lucas bring back Hamill]], where the smaller sentence is called a sentential complement to the main verb "consider".

The 3 choices for complement type in English are that-clause, for-to, poss-ing. (I hope I didn't forget any.) (1) [Lucas consider [that Lucas will bring back Hamill]], (2) [Lucas consider [for Lucas to bring back Hamill]], (3) [Lucas consider [Lucas's bringing back Hamill]]. So part of what you're asking, I take it, is what principles determine for a given main verb ("consider" in the example) which of the three complement types will be possible, and what nuances of meaning will be expressed by the choice.

I'm sorry to have to report that the answer is unknown, and it may well be that there is no answer. English speakers may have to learn ad hoc for each complement-taking verb which complements are possible and how to interpret them. So for an English language learner, in this particular regard, grammatical study is probably not useful, and lots of experience with interpreting and speaking or writing English is all that can help.

The name of the relevant field of study, following Noam Chomsky in Aspects of the Theory of Syntax, is "the problem of verb subcategorization". It was tackled by George Lakoff in his dissertation Irregularity in Syntax, and there is a very good more recent discussion in McCawley's The Syntactic Phenomena of English.