"If I had taken that bus I would be safe and warm right now."
This is correct. It's what's known in some EFL/ESL textbooks as a "mixed conditional" or "mixed 2/3 conditional" because the "if" clause has a verb in the past perfect (as in the third conditional) but the consequence clause has the construction "would"+bare infinitive (as in the second conditional).
Like a third conditional, the "if" clause describes a condition that is in the past - a past unreal condition that was not met. But like a second conditional, the consequence clause describes a consequence that is in the present (not a consequence in the past).
In the third conditional you'd say "If I had taken that bus I would have been safe and warm", but instead you want to express the fact that if you'd done things differently you'd be safe and warm now, so you use the mixed construction. It's perfectly standard and perfectly logical.
"If I die tomorrow I’d be alright because I believe…"
This, on the other hand, is badly worded. It should be either "If I die tomorrow, I'll be all right..." (first conditional) or "If I died tomorrow, I'd be all right..." (second conditional). (The form "If I were to die tomorrow, I'd be all right..." is also fine and is a variant of the second conditional. I use the spelling "all right" because in British English the spelling "alright" is sometimes considered nonstandard.)