Witches & War Rigs

I'm a big proponent of saying "you succeed without a roll because you have relevant expertise and there's no serious pressure", so that's a good move. 

Handy is basically a more limited version of Jack of All Trades. Handy lets you add another practical skill in which you use your hands (mechanic, carpenter, etc), but Jack would let you add any vocation. Jack is an opportunity to have your armored juggernaut warrior say "Yes, I've read a book or two. Before I joined the army, I actually went to medical school."

There's nothing wrong with your reading skills. In an older version of the layout, the war rigs were on the vehicle chart but I moved them to the armor section to emphasize that it's a suit that you wear and not a vehicle that you drive. Learning to move and fight normally in a rig is more a matter of athletic adjustment than learning a piloting skill. 

I'll be fixing some errata and uploading a corrected version of the book soon. Standard rigs have a fuel tank that holds 12 slots of fuel, like the ruggedized motorcycle. The Warrior starting packages should also state that any vehicles or rigs start with half a tank. The vehicles & travel section on page 52 still includes the fuel expenditure of rigs: they're very efficient, but don't carry much with them. If you have a player that likes to count all the copper pieces, they can have some logistical fun in figuring out when it's better to make the war rigs walk or ride in the back of a truck.

The "cargo box" war rig mod is big enough to hold a 5-gallon jerry can and I would allow a player to use it as an extended fuel tank by doubling the cost to account for extra fuel lines and such. (Or they could just keep a jerry can in there and get out of the rig to refuel.)

There usually aren't contested rolls. Instead of "PC rolls to sneak and tries to do better than the NPC roll to spot", it's just "PC rolls to see if they can sneak past a guard".

It's up to you if you want to add extra difficulty modifiers to the PC roll. Outside of combat, I usually just use 5 difficulty levels:

1. Easy enough that you can do it without a roll (because you have suitable skills, equipment, or circumstances in your favor).
2. Easy, but not guaranteed, so roll with advantage.
3. Normal
4. Hard, so roll with disadvantage.
5. Too hard. "You can't get there from here" so try something else.

For sneaking specifically, some people might ask "Shouldn't it be harder to sneak up on soldiers who are on guard duty than just a normal person?" My reply is sneaking past guards is the normal use case for sneaking, so it's just a normal roll with advantage or disadvantage for anything really unusual. 

Disadvantages would be things such as a monster with unusually keen senses or guards after an alarm has been raised so they're actively looking for you. I don't worry too much about the difference in difficulty of sneaking up on NPCs with Wisdom 11 vs 14. If you want difficulty ratings, you could just use NPC Ability -10 as the difficulty (so you would need a successful roll higher than 4 to sneak past the Guard with a 14).

For advantages, I like to leave it up to the players to think of ways to tip things in their favor. Some things I've seen include timing the guard patrols to pick the best moment to run for the fence instead of just trying to jump from shadow to shadow like a ninja, throwing meat as a distraction for guard animals, and using a slingshot to throw a pebble to make a noise for guards to investigate "over there".