Well, I'm very rusty but I think the parts are way easier if you stick to some pattern, at least for the dogs. The advice I'm giving you is from the top of my head, so not saying this is the best way.
Is your DS level 2 or 3? Because lvl 3 will give you the 360 UT which is very useful on this part.
For the dogs, presuming you have a DS lvl 2:
Never commit your combo's to much. XYX, bbb, OL Y (hold) and release (without ninpo) makes Ryu slash forward 'through' the enemy. That works pretty well OL. Then dash-jump shuriken repeat and when there's essence, do an ET/UT. You can sometimes commit and do something like XYXXXX, or a FS but the dogs tend to be unforgiving.
I also believe ->XX.... works to some extent (the rolling slash). Block/dash always everywhere.
DS lvl 3: 360UT, dash/jump. bbb, 360 UT etc.
Oh, and wall cancelling is a very important and useful shuriken ability, but sometimes it can throw you off if you don't expect to land on the ground but against the wall, so try to spam less and be conscious about when and where.
The archers are really not that hard once you find your rhythm. Start running on the water and stay as most right as possible. Then immediately when you're about to run past that wooden bridge pole that makes them able to target you, do a FS to the right and you'll land on the platform. Usually you can follow up with a GT. Once on the platform, the rest is easy. Stay mobile and use i-frame moves. Decide if you really need to heal, or save it for the LotTG bridge fight. (there's also blue essence in the target practice area)
For the FT room: Dash/jump (windrun) GT->OT, XXYbY, ET/UT and counterattacks are the way to go. Again the ->XX rolling attack can be great too, but it takes more practice. Try to maintain your overview and try to pay attention to when the shurikens in your body are about to explode. You can also mix things up here, i think that's your best bet, to not get predictable. FS, ID etc. If you do an ID, always be ready to insta counter OL or Block/Dash away. And try to single out the IS ninjas. But don't ID when everything is still alive and kicking.
If you find the moment you can do a wall attack spin into manual combo but i totally forgot the buttons for this. When done right, you'll probably delimb several enemies at once. It can be useful to 'save' a delimbed ninja to have an i-frame backup, but this is risky.
You should be able to upgrade your DS to lvl 3 before entering the FT fight(if you want that) DS 360 UT makes things slightly easier, IMO. But in general LS lvl 2 and FT lvl2 makes this level easier, however, if you're comfortable with the DS, go ahead. The essence you earn from the the lvl 2 LS 360 UT is very useful though.
IIRC, you can have a lvl 2 DS and LS before the FT fight with enough essence to also upgrade the FT to lvl 2 before the LottG fight and maybe even have a Lvl 3 DS and lvl 2 LS (or vice versa) before the FT fight.
Have fun :)
If you always do what you always did, you'll always get what you always got.