it would not hurt to call, even if it ends up as a voice message, and ask something along the lines of
" Looking forward to startig work on ________ and was wondering if there was anything I should bear in mind for the first day"

Even Human Resources could be an effective place to call...

It's a small business, sort of a startup, and there is a Controller who seems to run the office so I was planning on calling her again.

We already had a nice email exchange last week, so I'll call her again next week. I don't start until June 10th!

When I wrote them all my thank emails, I did mentioned they could contact me for any reason in the meantime, but no word from my actual bosses. I guess they aren't going to bother talking to me until I show up.

So if you do feel need to call, I would be very low key about it, keep it short.

I got hired years ago in face to face interview, salary was agreed upon, I went to work, payday came 2 weeks later, my check was short, so I asked the director who hired me about the "mistake" and she looked me in the eye and said "I decided the position was not worth the price we agreed upon. You can quit, if you want to".
I could not quit, I needed the job.
But I DID end up using my considerable talents to make my self indispensable, got put in charge of a new grant funded program, and looked around for a better job, which I got within a year, using the current job on my resume.
New job hired me, I asked for a confirmation in writing, they were a large corp. and a bit put out that I asked, but sent me the letter, which included the agreement to move my household from Fla. to Mobile,Ala. for the new job. and confirmed the
salary offer, ( which was 6,000 a year increase over my old job.)
Then I waited until a full staff meeting, stood up and announced my resignation, then handed the written resignation to the director who had lied to me.
She spluttered and then blurted out " I don't like people who quit when we need them"
and I replied " I don't like people who lie about hiring salaries" and walked out.
( the whole staff hated her and knew about the salary trick)

You gotta take care of yourself esp. in these times, and the email exchanges seem to have done that, esp. if they quoted a starting date.
Make a good impression and show up 20 minutes early.

Actually, this job is contract to hire, so after 60 days I find out what my real salary is. All I can do is trust that it's in the same range as the initial rate. I guess it depends on how good I am the first 60 days. My field doing great now, recruiters calling me all the time, so I wouldn't have trouble getting another job if necessary (besides explaining why I left so soon). I doubt it will come to that. Your story is inspiring and gives me courage.

...it shouldn't be a problem. If you have concerns, check-in with their HR mid-week and ask about anything they may need from you on your first day and what time to show up. Check with the recruiter whether they need a drug test, background check or other preliminaries. It may expedite your first day if you ask about any required agreements (IPA, pubs & patents...). Make sure your ID is current and your SS card is readable.

Have some good time off next week.

Since the recruiter gets paid for the referral I'm sure the company wouldn't want to waste that money they spent. I will contact them midweek, like you said, it's what I was planning. Weird they haven't asked for any info, drivers license or anything. It's a small business but they are in the city and I think they are very reputable, just really casual I guess.

...you will be an employee of the recruiter's shop. In January they will send you a W-2 or a 1099 depending on your arrangement. Since the client company won't be your employer for a few months, the ID and I-9 burden is on the recruiter's shop. The client may need intellectual property agreements. If they're a small business, they may not have concerns with those issues.