Help with train tickets

Hello! I will be visiting Japan for the first time next week and I need some help buying train tickets. I've left everything to the last minute, and I am trying to figure out my transport options, but despite the myriad posts on the JR pass and all the ticket options, I'm not sure what I should do.
I will be arriving at Haneda aiport, I've booked a hotel in Tokyo which, from what I understand, is on the Keikyū Airport Line. I will stay in Tokyo for one day and then I will go to Nagoya (using the Shinkansen). After a 5-day stay in Nagoya (where I will do some light travelling in the city), I will be going back to Tokyo for a two-day stay, and back to Haneda (I will be staying the same hotel). The whole trip lasts nine days: 22nd to 30th August. My questions are a) do I need the JR pass? If I buy the 7-day pass, it doesn't cover the last two days of my trip and it seems to me quite expensive, b) should I buy a IC card instead? I am not sure how to do that, c) should I skip booking any tickets overall, and just buy them at the train stations or is that too risky?
Many thanks!

a) do I need the JR pass?

No, because 7 day JR pass is more expensive than the round trip ticket from Tokyo to Nagoya.

b) should I buy a IC card instead? I am not sure how to do that

Yes. Follow the Video instruction.

https://www.youtube.com/watch?v=TSPISZ8X6kY
https://www.youtube.com/watch?v=nU5bZIGVeAo

c) should I skip booking any tickets overall, and just buy them at the train stations or is that too risky?

You can just buy Shinkansen ticket at the train station from the ticket vending machine when you do the trip.

First you must find appropriate vending machine, which is

Reserved Seat Ticket Vending Machine.

https://www.jreast.co.jp/multi/en/ticket/purchase.html

https://www.youtube.com/watch?v=IpSMSn0DzgA&t=210s

Hi, I think use jordain to plan, buy tickets on the spot will be easy. But I havent used a pass since our first trip, and I've never tried icoocca or whatever

In addition to Frog's comments - you can also buy a shinkansen ticket from a manned ticket office if you find the machines to be a little confusing. There are lots of trains so don't worry about booking before you arrive in Japan. You can always buy your round trip shinkansen tickets on your first day there.

Thank you everyone for your very helpful comments! Do you also know if I can pay at the vending machine using a credit card?

It's written in the link provided by frog1954:
https://www.jreast.co.jp/multi/en/ticket/purchase.html

Reserved Seat Ticket Vending Machine -> Credit cards can also be used.
(With a picture of what that machine looks like, so you won't confuse it with the other machines that don't accept credit cards.)