I created an event and my wife and I tried but were not able to book a time slot. We tried with a sats cost and without. Does this part work yet?
I'm finally ready to share a project I've been working on the last few weeks, https://nostrcal.com
Nostrcal is my shot at building a calendar management system for nostr that I'd want to use. It leverages gift wrapped nostr calendar events to be one of the first calendars on nostr that stores your events privately for only you, and your assigned participants, to see.
I've also created a calendar management tool similar to Calend.ly that allows users to book time on your nostr calendar. The tool allows you to create Event Slots that define how you'd like to accept meetings and display availability. This gets stored as a 31926 list and converted into a naddr link that you can then share with people you'd like to book time with you.
Private events are stored to your kind 10050 relays. To help test I've built my own relay that supports storing your wrapped nostr events and NIP-52 based events at relay.nostrcal.com. It's currently being offered as a whitelist relay for 1000 sats per 30 days. You can use any existing DM relay that supports storing wrapped messages though with no problem!
Since private events hide their details from everyone I'm using a new 31927 event kind to create busy slots. These busy slots are public events but only show a start and end time for being busy. These are intended to be published when users create private NIP-52 events so that time management can be handled by the booking system still. If you don't want to publish the 31927 to ensure complete privacy, you can disable that on the event creation.
There's still support for public events as well. You can look over the wide range of nostr public events to find interesting things you'd want to attend. This area is still very basic but I plan to add better event filtering and sorting in the future.
My modified NIP-52 is available now at https://github.com/patrickulrich/nips/blob/master/52.md. I haven't submitted this as a PR yet mainly because I've been improving it as I think through flows. I think I'm getting close to the point I feel good about the NIP revisions and will submit it soon.
I'll also be releasing all the code for the site soon. Most all of the code was written by my friend Claude with me only stepping in when Claude must have had too many drinks and would go haywire. There's still bugs and memories of those haywire moments that I'd like to get cleaned up this weekend but will be posting it soon for anyone interested in looking under the hood.
I've tried to make sure most things were working but I constantly keep finding new bugs that need squashed. If you give it a test and see any issues please share them on this post. I'd love to continue improving the system to make it as user friendly as possible. I have a ton of new feature ideas like ics import and syncing with other calendar services to import your existing calendars into nostr events planned but want to get the base working as well as possible first.
Discussion
Sats cost isn't implemented in any form yet so it shouldn't be a problem either way. Do time slots populate or do you never see any times available?
Time slots do populate. I got one of my coworkers to create an account and then she could take a time slot. So maybe my wife was not signed in.
I see that if someone creates a key on the site, I see they get a prompt to download their key. I don't see a way currently to get your key if you miss that prompt, correct?
I missed this somehow so sorry for the delay in getting back. Right now I don't have any way to export it past the initial time but I'll look to see if that's something I could add to the settings page.
Do you see something like this on your booking link or are the times never showing up? Also what sign in method are you using? 
I get past that point. It's in the scheduling page where I need to enter a name, email and comment. The Book Event button was greyed out.
When I just checked this morning it's now clickable.
Initially I had set it to 21 sats, so if payments don't work it makes sense. I then changed it to 0 sat and it was still greyed out. Maybe it was just a delay in the relays updating the event to have no fee?
Ahh, got a few work buddies to try and book without having a nostr account, so they probably just need to be logged in to click submit for a time slot. When I did it this morning, I was logged into my account and I was able to book a time.
I think I have a fix coming in the next update push for this. If a user tries to book but isn't signed in the button will open the 'Login' modal to let them either Sign Up/Login and complete the booking. It should def help reduce that confusion point. Sat bookings still to come.