@Sean95
I have a Yale Assure SL Deadbolt with August Connect. Mine does not have the key option but there are other versions that have it. If you're thinking about Auguest, I definitely recommend going with an established Lock manufacturer with the August Connect module versus buying one of August's Locks. Here is how mine fulfills your criteria.
a) Send me an alert if door is left open (or ajar) more than X amount of time.
I can setup Smart Alerts within the August App to notify me if the door is left ajar any time it occurs or within a time range. There doesn’t seem to be an option to alert me if it’s left ajar for X minutes, however I can always look in the August app manually and see how long the door has been left open.
b) Automatically lock if and only if the door is closed.
I haven’t set this up but according to this article, it can be done with the Doorsense - https://support.august.com/intelligent-auto-lock-with-doorsensetm-S1mg1DI1A_f - However, I'm not clear on whether or not that requires a bluetooth connection and/or if that's only for the August Lock or also an option in the Yale. I can check for you later today when I'm at my AirBnb. Will report back.
c) Option to control with a keypad (so not everyone is forced to download an app)
Absolutely necessary and my lock automatically sends each guest a code which activates/deactivates at check-in / check-out. I can also manually adjust these times.My cleaner has her own permanent code as well. I have a backup, emergency key on the property as well, if they get locked out, which will open the other door and can also unlock the doorknob on this door. There is absolutely no way a guest can get locked out. Guests can also download the app but that's optional and not necessary.
d) Most economic option possible. - It’s not the cheapest but you can find good deals on eBay, OfferUp or FB Marketplace. I waited until I found a good deal and then pulled the trigger. You just have to check often because they don’t last long.
I've been mostly happy with the lock other than a few minor hiccups. It does require a hub and my only complaint is having to replace the batteries every few months. The lock doesn't work right with rechargeable batteries either, so keep that in mind. My plan was to use rechargeable batteries but that's not an option, unfortunately. The rechargeable ones don't last as long and you don't get the notification when they are running low. So you have to use regular alkaline batteries.
I know you don't care about home automation but there may also be another option to ONLY lock the door if it's closed, by using Homekit if you have an iPad or AppleTV in there. I'd have to do some additional research.
In any case, I will check about number 2 and post back here later this evening.