Goodness! I'm trapped in a conundrum, and I don't even know it!
None of the "would" + base-form-of-verb patterns are past. They are all conditional.
"would like" is an idiom for "want"; the substitute is often made for politeness.
"I would go" and "That would be nice" are pure (non-idiomatic) conditional, but their corresponding "if" clauses are not explicitly stated.
"He would think that" is another idiomatic formula in one reading (stressed "would" suggesting something like typical behavior) and a pure conditional in another reading (unstressed "would") - again without an explicit "if" clause.
Historically "would" was the past of "will". That's why in modern English the future of the past (or "conditional") is formed with "would" (or "was going to") and the future of the present is formed with "will" (or "is going to").
As much as it would simplify the structure of English to construe all modals as tenseless or to construe them all as tensed in pairs like "can"-"could", "shall"-"should", etc., neither "all-or-nothing" theory works very well. I think we need to allow ourselves the scope to explain some structures in terms of one formulation and others, in another.
Further, it is useful to separate systemic and non-systemic uses of modals. (The idiomatic uses mentioned are what I'd call examples of non-systemic uses.)