Is there a way to make it to where I can have different initial spawns in different game types on the same map, and have all the initial spawns show up in the map when doing "Basic Editing"?
Indeed there is, if you wish to spawn a certain place for say infection, rather than the normal spawn areas, you would first drop a new initial spawn point. Next you would go to the advanced options of that spawn, and set it to "Gametype Specific: True" and then give it a label of "Infection" (the label has to be done from editing in that gametype). This will create a initial spawn that will only appear while playing infection on the map, and the same principle can be used for any game type.
Duck is correct. You want to make sure that each initial spawn point has a unique label. If you leave any initial spawn point label-less, then it can be used by all game types. Another point to consider is that to make your map more maintainable (easier to make changes to), you might consider having only one initial spawn point per game type per team (e.g., one red initial spawn point for infection, one for slayer, one for flag, etc.) and then surround them with neutral respawn points. As long as they are far enough away from each other, the single initial spawn point for any game type will be the place for the first team member to spawn. The next will spawn as close as possible using a neutral spawn point. The next as close to them as possible, and so forth. This is very convenient, but it is entirely up to you and the layout of the map.