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.
what kind of wire do I need to run if my room is around 48 feet in perimeter?
Check out http://www.calculator.net/voltage-drop-calculator.html. Plug in numbers until you get an acceptable value. For example, 14 AWG copper at 12 V and 2 A load (1.5 watts/ft * 16 ft / 12 volts = 2 amps) gives you 11.52 V at the end. If that's within spec for your LEDs then 14 AWG copper it is.
And will running it in parallel help with the dim at the end?
Yes. With 4 strips in series the total voltage drop across all 4 is 12V, so each of the 4 necessarily sees a much smaller drop (average 3V each because 12V / 4 strips = 3 V/strip). In parallel they would each see 12V and the limiting factor becomes how much total current the power supply can provide.
And finally do I need to get a different power supply if I change it?
Doesn't sound like it. Worst case your four strips require 64 feet * 1.5 watts/ft / 12 volts = 8 amps. Your power supply gives you 25% extra padding. Should be more than enough.
Best Answer
The added up size of the breakers is not equal to (generally it's much greater than) the actual load.
There are code guidelines, or you can look at what's expected to be powered and what its actual amp draw is.
Tankless electric water heaters are foolishness, and the huge amperage requirement is one reason why (the lack of power savings is the other main one.)
Also you are apparently in 120V/240V land and don't realize that 2 20A loads on opposite phases at 120V are the same as one 20A load at 240V. That alone gets you down to 175A on your current list.
You also need a minimum of 2 20A circuits for kitchen outlets, and it's generally advisable to have at least one other circuit (15A is normally plenty) for the refrigerator and nothing else.
For a start, look at the actual running amps of your mini-split, .vs. its 40A breaker size. If you insist on the tankless foolishness, it's actual amp draw .vs. breaker size will matter as well, but it won't spare you much in that case. FWIW, that's one HECK of a breaker for a minisplit on a 288 sf cabin - I'm doing 1000 well insulated square feet with one that asks for 15 and uses maybe half of that.