Intiative will be affected, but AC will not.

• Initiative is DEX + d20. If your DEX is -1 then it's going to be d20-1.

• AC in your case ignores DEX because you're wearing heavy armor. You would take a penalty in either light or medium armor.

Reading the sections on both of these, they do not mention exceptions for negative DEX.

Yes. Any character who is proficient with a particular type of armor can cast spells while wearing that type of armor with no penalty.

There are two primary reasons for this.

1. There are better ways for mages to get a decent Armor Class. For example, Draconic Sorcerers get permanent Mage Armor for free at 1st level, which provides 13 + Dexterity Modifier AC. That's equivalent AC to a Chain Shirt (except the Chain Shirt will only let you apply a maximum Dexterity bonus of +2), and the Chain Shirt costs 50 gp and weighs 20 pounds. If you take a mage with high Dexterity, then it only gets worse as the armor gets heavier. In addition, any mundane armor heavier than 20 pounds will give you Disadvantage on Stealth checks, and truly heavy armor won't let you apply your Dexterity Modifier at all. Chain, Splint, and Plate Mail all have Strength requirements as well.
2. Arcane Spell Failure was a complicated and pointless mechanic. It only served to confuse newbies and slow down the pace of gameplay. Furthermore, the only spellcasters who have access to metamagic (which, aside from magic gear, was the primary way ASF was mitigated) are Sorcerers, which is also the class that needs heavy armor the least. Magic gear is much harder to come by now, so the likelihood of you finding a magic suit of armor made to be light and maneuverable is much more remote, thus removing yet another way of mitigating ASF.