Actually, it differs depending on the conjunctive particle.
You can also use the polite form before から, so there is no problem with 病気ですから or 降っていますから in your examples. However, only plane forms are usually used for some other particles, for instance のに, thus, 雨が降っていますのに、散歩します sounds odd (not ungrammatical but almost never used). This is because のに has a strong subordination to the main clause.
On the other hand, you need to unify the level of politeness for が (e.g. 雨が降っていますが、散歩します, not 雨が降っているが、散歩します), since these two clauses are closer to two independent sentences. Speaking strictly, this is more likely a coordinate clause, not a subordinate clause, though.
Incidentally, the tense in Japanese grammar is rather different from the English one. You'll come across it soon.