Am I under some kind of legal obligation to tell my landlords that I'm no longer a valid visa holder? I know I have to have someone look after my flat (can't leave it without a person for more than 20 days). I'll just have a friend house-sit. However, I'm hoping I can just NOT tell my landlord and leave all my stuff there.
If you do not have a valid visa, then your landlord will be breaking the law by having you live in the property. I'm not sure about what happens if you aren't physically in the UK but you are still renting the property without a valid visa.
I would be honest with your landlord and see what they wish to do about your tenancy. I would think they would be worried about what happens if your visa is refused and you cannot enter the UK again to pack up your things.
Also, if you are renting, you can't just have a friend house-sit for you unless your tenancy agreement states you are allowed to let other people live in the property... so you would likely need to ask your landlord for permission to let your friend stay there while you are gone.
If a landlord is found to be renting to someone without a valid visa, they can be fined £3,000.
The landlord is legally required to do a 'follow-up check' either 12 months after the tenancy agreement starts or just before your visa expires. They can be fined if they continue to rent to you after your visa has expired. If you fail the follow-up check and the landlord does not tell the Home Office about it, the landlord can be fined or sentenced to 5 years in prison.
See:
https://www.gov.uk/check-tenant-right-to-rent-documents/further-checksAlso, side question, if my Entrepreneur Visa took longer than expected to obtain how long before I could re-enter the UK (with a travelers visa?) as a traveler just to get my stuff in order? Like pack it up and ship it if I had to? (also if they denied the visa for any reason, hope for the best and plan for the worst!)
Well, UKVI would have your passport while the visa is being processed so you wouldn't be able to travel until it has been processed.
If you don't get the visa, you would likely need to then apply for a visitor visa to be able to go back (due to having a previous refusal), but that may also be refused if you have just been refused an entrepreneur visa, since it may look like you are not a genuine visitor.
Honestly, the safest thing to do would be to pack up and leave the apartment now, ship everything back to the US, and then start over when you have your new visa.
Tier 1 Entrepreneur visas are not often granted - only about 60 visas per year are granted in the US. About 2,000 Tier 1 Entrepreneur applications are made per year worldwide, and only about 45% of those are approved, with 55% being refused.
In comparison, for Tier 2 General visas, approximately 2,500 are granted in the US each year, and about 16,000 are granted worldwide. About 96% are approved and only 4% refused.