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.
You don't need to upgrade the supply breaker.
The sub-panel can be any size 50 amps or up. Sub-panel "size" is more about space than amperage, at any point past "at least as big as the supply" and space is a good thing to have which normally does not cost much.
It's often worth comparing the price of basic "main" panels that move in volume (100A, 200A) to the cost of a sub-panel which are sometimes MORE expensive, because they sell a lot less of them, or they cost a trivial amount less.
It's perfectly fine to feed a 100A(or whatever) main breaker from a 50 A breaker - the 100A breaker will be serving as a disconnect switch, and you will remove the bonding screw in that panel so it IS a sub-panel. Then you'll have plenty of space for anything.
In most cases the best way to add a wire in existing conduit (unless someone left you a pull string, and sometimes even then) is to pull out the wires in it now (and use them to pull in a pulling rope as you do), add the new wire, and pull all the wires back in together. Green insulated may be a better choice than bare for your grounding wire; Among other considerations, stranded pulls a lot easier than solid.
In the new sub-panel with a 4-wire feed, you'll keep the grounds and neutrals isolated (removal of the bond screw and using a separate bar for grounding connections) and you will either tie the grounds into the grounding electrode system the shed (as a separate structure) should already have had, or add that and bond it to your ground as well.
Best Answer
Alien alert!
That 200A Siemens QN(R) breaker visible in the center of the photo has no business whatsoever being in a Homeline panel such as yours. You'll need to replace it with a HOM2200BB and rearrange the other breakers so that you have four free spaces on the bottom right for it. This also gives you a LK225AN you can mount on the lower right-hand neutral bar so that you can cut the neutral wire (aka the black wire with the yellow stripe) and re-strip it to get a fresh set of strands, then fit it into the new lug.
While we're dealing with breakers, this panel has one too many handle throws in it at the moment; that's trivially dealt with by replacing one of the top pairs (left or right) of HOM120s with a HOM220.
As to other matters...
I noticed that you have several NM cables that appear to be exiting this outdoor panel and heading somewhere that isn't immediately inside. Those cables will fail on you due to the paper wrappers wicking water, so it's best to replace them with either UF cables if they're running free, or with THHN wires if they're in a conduit instead. (There's also what appears to be a UF cable in one of the conduits that should go as well, as that conduit's likely overstuffed as a result.)
Atop that, it appears that whoever installed this used a PVC plumbing pipe for electrical duty, shoving the end of the pipe through the hole instead of using a proper adapter fitting while they were at it. This is no good at all, and needs to be torn out and replaced with a proper conduit run with proper fittings.
Finally, the surge protection device(?) attached to the right side of the panel needs to be examined for proper watertight installation, as I don't know what is going on there, and the conduit entering the box bottom center needs to be fitted with a split conduit bushing of the correct size so that the wires don't abrade against the edge of the conduit.