1. For beam support, you need at least two, possibly three, studs to support the beam on either end. Check with your local code enforcement official for specifics in your area. Minimum support area on top of the studs is also a local code issue. Allowable techniques for connecting the beam to the studs is also a local code issue.
2. You do not need a vertical support in the middle of the beam PROVIDED you properly size the beam. Since there are typically no tables or simple formulas for sizing a beam, you may wish to hire an engineer, architect or qualified contractor to size the beam for you. If you purchase a glulam or other engineered lumber beam, the lumberyard that supplies the beam may have a resident engineer available, and they can include the cost of design in the price of the beam.
3. The studs (you refer to them as trimmers) do not need to extend all the way to the basement. They can be extended to a major support beam at the next lower level, HOWEVER it is critical that the strength of the support beam be analyzed by your designer, as it is possible to overload the support beam by adding point loads to it, which could lead to catastrophic failure. The connection detail between the studs and the support beam is typically driven by seismic considerations, so your location is crucial to the required connection technique, your engineer will understand this issue.
4. Should you elect to extend the studs all the way to the basement, you need to check buckling in the studs (your engineer will understand how to do this complex analysis). You also need to construct an adequate foundation for the studs. The design of the foundation is typically driven by the load to be supported, the size of the column, and the bearing capacity of the soil. Your engineer will design the footing for you. Local code is likely to dictate a minimum size for the footing.