This is sort of a trick question. There are several valid answers depending on the given conditions.
In all cases SEE and AVOID is RULE 1
Conditions: (Apologies to the FAA if I do not use the term NON TOWERED)
1.Airport is IMC and controlled
2a.Airport is IMC/not controlled but 1 mile clear of clouds and appropriate ceiling
2b.Airport is IMC/not controlled and less than 1 mile clear of clouds
3.Airport is VFR and Controlled
4.Airport is VFR and uncontrolled.
Conditions 1 and 3 The pattern is what ever the tower instructs.
Condition 2a. Aircraft are conducting instrument approaches and are on an instrument flight plan. If ATC has RADAR coverage, they will advise the IFR aircraft about the VFR traffic. The standard right of way rules apply. The IFR arrival does not have priority. RULE 1 applies. This is always something that pilots argue about. But ties goes to the NTSB for accident analysis.
Condition 2b. There should not be any VFR touch and goes. Subtle distinction, but IFR traffic may be cleared for the option and the option may be a touch and go. I would not expect ATC to clear 2 A/C for the option at the same time.
Condition 4. Right of way rules and RULE 1. IFR Aircraft may utilize the straight in approach, however, they may be required to give way to aircraft established in the pattern. If they can sequence themselves in fine, otherwise they join the pattern. Generally, a sidestep to an upwind entry works.( But not if there is glider or helicopter traffic).
I will let someone else play with this a bit.