Yes, you can order referencing on any potential tenants. All you need is their permission, their phone number and their email address. You can begin that here: OpenRent Standalone Referencing.
OpenRent is designed for landlords to let homes in the Private Rented Sector and assumes an Assured Shorthold Tenancy in England & Wales, and a Private Residential Tenancy in Scotland.
I don’t think that an AST is the appropriate agreement for short term lets.
One of the basic premises of an AST is that it is not possible to evict the tenant before six months, unless grounds for Section 8 exit, and even then it’s very unlikely you could complete the S8 process before six months: and certainly not at three months. Therefore, it’s not possible, legally and practically, to use an AST for a three, four or five month tenancy.
You could use it to offer a six month tenancy, but this would give the tenants rights that you would not normally want to give to short-term renters. E.g. if they decided not to leave, it would be very hard to repossess the property.
To answer your final question, we do secure tenancy deposits on behalf of landlords when they use our tenancy creation service, Rent Now. But Rent Now requires an AST.
Hope this has been helpful.