In 'I'd probably have a better position by now', is 'would probably have' Conditional mood or Past tense?
"Would" is a modal verb, "have" a bare infinitive. Technically "would" is the past tense of "will". "Would" has various uses, including expressing a conditional (i.e. the result of a particular condition) and expressing the future in the past. Here "I would have" is conditional - it expresses the results that would follow if a certain condition were met.
Some grammars refer to "would have" as the conditional form of the verb or as the conditional "tense". But most grammarians today would not regard English as having a conditional tense nor a conditional mood.
In 'if I did', is 'did' Conditional mood or Past tense?
"Did" is clearly past tense, but it is not talking about the past. Rather, it is talking about a hypothetical. We don't, however, use the term "conditional" for this use of "did" (since the term "conditional" is usually reserved either for verbs with "would" or for conditional sentences).
Some would call this usage the "past subjunctive", but others would argue that the term "subjunctive" is irrelevant here because there is never any indicative/subjunctive distinction in the English past tense - the sole exception (not universally adhered to) being the subjunctive-like use of "were" in the 3rd person singular. As this is the only instance where a distinction can be drawn, it is better regarded as an isolated relic, sometimes dubbed the "irrealis" form..