I've labeled your image, to help you understand what's going on.
Off to the left, the grounding electrode conductor enters the box and terminates at the grounding bar. The feeder coming in the top of the disconnect has three wires, two ungrounded (hot) conductors, and a grounded (neutral) conductor. The two ungrounded (hot) conductors terminate at the disconnect, as they should. The grounded (neutral) conductor terminates at the grounding bar, as it should if this is where the service is grounded.
The feeder leaving the bottom of the panel has two ungrounded (hot) conductors, which terminate at the disconnect as expected. The grounding conductor terminates at the grounding bar, as it should. And the grounded (neutral) conductor terminates at the neutral bar. Unfortunately, since this appears to be where the service is grounded, the grounded (neutral) conductor from the lower feeder should be connected to the grounding bar.
As it's wired now, the grounded (neutral) from the lower feeder is connected to an isolated neutral bar. Which means that the grounded (neutral) conductor feeding the panel is floating, or not electrically connected to ground. Without a reference to ground, the voltage potential between either of the ungrounded (hot) conductors and the neutral can be anywhere between 0 - 240 volts.
Solution
Service grounding location
If the disconnect panel is where the service is grounded, you should move the white wire from the lower feeder to the grounding bar. Or you could bond the neutral bar to the grounding bar, using an appropriately sized conductor.
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Not service grounding location
If this is not where the service is grounded, you should move the bare conductor from the upper feeder to the neutral bar.
Click for larger image
Based on what you've said in comments; and because it's a 3 wire feeder and not a 4, it appears that this is where the service is grounded.
That Federal Pacific panel has to go as they are dangerous. As far as concerns about your skill, you sound like that particular kind of newbie who is well capable of learning to do it all safely and well; however my hunch is you are still thinking too much, and need to read a little more. It's OK, we all start there.
Normally, just replacing a sub panel is a straightforward thing. Change panel, reattach wires, done. However, this is only part of a project with a much larger scope. You must contemplate (i.e. ask your permitting authority) whether you have crossed the legal threshold of a remodel. If you have, everything in-scope must be done to all current codes. Even ADA!
The 2-circuit requirement for kitchen outlets is not intended to mean "1 circuit for this wall, 1 circuit for the other wall". I'd encourage some more research but if it was me, I'd interleave the outlets, every other outlet on a different circuit. Also there's nothing wrong with more than 2 outlet circuits in a kitchen, the whole point is to prevent trips when the chef is madly at work, so the chef isn't hobbled with limitations like having to put the toaster here and the George Foreman over there, and avert ugly workarounds like extension cords draped across sinks or stoves, etc.
I see you plan to go 12 AWG wire for almost everything (that's what 20A breakers mean) - that's awesome. Feel free to kick the refrigerator and smoke detector up to 12AWG also - that way you don't have to buy any 14AWG wire. I don't own any! If you have some other reason to use a 15A breaker you are welcome to use that on a circuit wired in 12AWG.
You may want to run the water heater circuit in 10/2 or even 8/2. That will allow you to easily upgrade to an electric water heater in the future. Still use a 15A or 20A breaker because the outlet is still only good for 20A. There is a trick to fitting 8+ AWG wire on a 15-20A outlet, just ask.
I would go with a much larger panel. You have either -1 or 3 circuits left, and that's too little headroom for my comfort. Getting a larger panel is dirt cheap compared to the cost of replacing perfectly good breakers with duplex breakers (I call them double-stuff) merely to shoehorn everything in. Also, larger panels in combo-packs come with more breakers and that is far-and-away the cheapest way to buy breakers. Another reason to avoid duplex breakers is if you ever need AFCI, GFCI or whatever future thing comes out - those are much more expensive in duplex breakers because of the miniaturization required. Don't think you must use a 100A panel - you can use a larger panel (200A), you just can't use a smaller one (70A).
Remember each sub-panel must have its neutral bus bar separate and isolated from its ground bar. That means removing bond straps, magic green screws, neutral bar kits, whatever the panel requires to do that. You might consider a panel with a neutral and ground bar on each side of the panel. That's a convenient feature so wires don't have to cross over the panel.
Keep in mind how your house got a dangerous Federal Pacific panel. The last guy bought cheap. Feel free to research the good-better-best that each manufacturer offers, the price differential for "best" is quite small compared to the overall cost of a remodel. You may find better selection and better prices at a real electrical supply house.
There is nothing wrong with more ground rods. Go nuts. The key is that all the grounds are connected to each other by wire - and they are not connected to neutral anywhere except one place - the main service panel.
Best Answer
There's a concept called grandfathering which says If the work was legal on the day it was installed, then when Code changes later, it's still legal. You're not required to tear out a bunch of wiring every 3 years when Code is revised.
This gets into a lot of lawyering about when the work was done and what was legal then. However we can safely pass judgment on things that were never legal.
There are at least 3 panels in this building. If you look on the "main lugs" above the breakers, you see 2 wires tapped onto those lugs.
Those wires are not service wires. A service is what comes in straight off the meter. Once it's past the main breaker, it is feeder. And both panels - this and the next - share that 100A of feeder. Daisy chaining panels like this is legal, except for...
The most obvious code violation is the double-tapping of those main lugs. Can't do it. Those 2 wires need to go into a 3-void Polaris connector, then a same-size pigtail needs to come down to the lugs so there is only 1 wire per lug.
Your subpanel doesn't have any ground bar, but that's not unusual when you have a panel that is connected entirely with metal conduit or armored cable, since that metal shell covers the need for grounds. I do see one ground wire that some epsilon-minus put on the neutral bar; that's not OK and must be dealt with.
To your questions:
#1. Does it meet current Code standards? It doesn't have to. It's grandfathered.
#2. Can I simply change a nonconforming panel? You wouldn't need to change the panel; unless you smuggled it in from a third world country, surely the panel is legal stock. You might have to correct defects, like the double-tapped lugs.
#3. Does NEC limit number of rooms per breaker? Not as you'd think. NEC 2020 has requirements for a number of dedicated circuits. 2 dedicated kitchen receptacle circuits. Dishwasher. Disposal. Built-in microwave. 1 dedicated bathroom recep circuit. 1 dedicated laundry room circuit. Almost all of them are GFCI or AFCI, and the AFCI must be done at the breaker, unless the route to the first receptacle is in metal conduit.
I don't understand why you're saying that the work has to come up to modern Code. It doesn't. The only exception is if the work was done illegally (i.e. without a permit), at which point the city is at liberty to make you tear out all the illegal work (even if there is nothing wrong with it) and then make you pull a permit to re-do the work properly. Since the permit would be this year, this year's Code would apply.
Lastly, as you have been cautioned, a landlord cannot DIY electrical on a rental unit. All work must be done by a licensed electrician. It really doesn't matter what your financial situation is; you just can't do it. That law is to prevent landlords from doing inferior work that puts others' lives in danger. The logic is an owner-occupant has incentive to do the job properly: a landlord only has incentive to do the work as cheaply as possible.