Here's one possible, simple, explanation, the starter relay might be in the stuck closed position, the temperature relay might also be in the stuck closed position.
Some workshop manuals have the relays incorrectly labelled, verify them against the wiring colours in the wiring diagrams.
To simplify the way forward, you could deal with the starter relay problem first, then look at the fan problem.
Removing the temperature relay will probably prevent the fan from working (this would also indicate that the fan problem is with the relay or upstream of it) Now check the starter relay, you should not have continuity between pins 30 and 87, if you do, that's a problem. A start attempt with a sad battery often results in a stuck relay. If you put a new battery in, it could still be stuck. If it has continuity, hold it two feet above a concrete slab, let it go, pick it up, it might now be fixed.
With the temperature sensing relay, you should not have continuity between pins 15 and A2, if you do, that's a problem.