If + should + verb is a form of type 1 conditional.
source: dictionary.cambridge.org/grammar/british-grammar/should
[excerpt]
We sometimes use should in hypothetical conditional clauses with if to express possibility. It is formal:
[information leaflet in a hotel room]
If you should wish to use the Internet, there is a code available at the reception desk.
If you should decide not to go on the trip, you will get a full refund.
[end excerpt]
Given examples:
1 - If it should rain tomorrow, I would stay at home.
This appears to be a nonstandard type 2 conditional.
Standard version: If it rained tomorrow, I would stay home.
2 - If it should rain tomorrow, I will stay at home.
This is standard and equivalent to the type 1 conditional "If it rains ...."
3 - If it should rain tomorrow, stay at home.
This is standard and equivalent to the type 1 conditional "If it rains ... stay at home." "Stay at home" is a command to the listener [essentially, "If it rains ... you must stay at home."]