Luggage space on public transport

I'll be visiting Japan for 15 days next month and plan to do a lot of travelling by train between 6 cities or so (including a 3 hour bus ride to Shirakawa-go from Nagoya). However I intend to do a bit of shopping in Tokyo (my last stop) and Osaka as well as picking up any local souvenirs that may catch my eye. Therefore, even though I'll be on the move quite a bit, I still want to bring a medium backpack (containing a small shoulder bag for daytrips) and a small/medium suitcase.

My question would be is this feasible on public transport? Can I keep both pieces of luggage near me (e.g. by my feet) on JR Haruka trains? Would there be enough space on the Nagoya to Shirakawa-go bus? Would it be hard to navigate the Tokyo/Osaka/Kyoto subway systems with luggage(suitcase) for a first time visitor?

Also, would it be possible for my hotel (a business hotel) to hold the suitcase for me for 2/3 days if I'll be returning to spend a couple more nights there? Or should I just look for large coin lockers? Can anyone confirm if they exist in Osaka/Umeda station (near my hotel)?

Can I keep both pieces of luggage near me (e.g. by my feet) on JR Haruka trains?
You may not be able to keep all the luggage by your feet but there's a luggage space on this train.
See this picture :
http://www.jr-odekake.net/train/haruka/accommo.html

Would there be enough space on the Nagoya to Shirakawa-go bus?
I guess yes.

Would it be hard to navigate the Tokyo/Osaka/Kyoto subway systems with luggage(suitcase) for a first time visitor?
Most stations have elevators/escalators but some not. And avoid morning rush hours (7:30-9:30) when you travel. You can hardly sqeeze yourself into the train even without luggage.

Also, would it be possible for my hotel (a business hotel) to hold the suitcase for me for 2/3 days
It depends on hotels. Ask them when you check in. If you're lucky, they keep your luggage. If not, then use coin lockers at the station.

if they exist in Osaka/Umeda station
Yes, there are. But large lockers are first to go. If lockers are full, find the luggage storage office (Tenimotsu Azukarijo 手荷物預かり所). Major stations, including Osaka have this office.