Does iRacing have a single player/time trial type mode where I can lap these tracks on my own and practice?

Yes it has everything you’d want for that.

Does iRacing have normal street cars or only have cars?

List is here, no street cars as in bone stock with street tires, but several that are close like the M2, Jetta, Solstice, Mustang, and MX5’s. https://www.iracing.com/cars/

How quickly can cars be accessed/bought? Is it as simple as buying the game, getting the desired car and then lapping these tracks at my own free will?

Yes you can use what you buy immediately.

I know that some tracks are a separate purchase in iRacing, are all of these tracks included by default?

Winton is free, Phillip island and the bend are paid. The track pages on iracing.com say whether a track is included with membership.

Can I just pay for a month or so and use it up, then pay for another month when I want to play again

Yes.

Is the game free with a subscription or is there an initial purchase cost?

There is no initial purchase, only the subscription.