As an example, Wondergirl from the latest Teen Titans set has this power:

WONDER GIRL'S LASSO: Wonder Girl may begin the game with Wonder Girl's Lasso assigned to her by paying the point cost. This object does not count toward your object total and is scored and removed from the game when Wonder Girl is KO'd.

Can you start her with her Lasso AND include a Resource on your team? Does the "not counting toward your object total" indicate it isn't regarded as on your force per se?

Of course not just her, other characters, Harvest et al, have a power like this. Does it work the same for all such characters?

In this specific example, it is pretty clear her Lasso is part of your force (just not your object total). You could not have it and a resource on your force.

If a character like this somehow made it clear the object was not part of your force, it would work with resources. All the ones I can think of won't work this way though.

Even though the object doesn't count towards your object total, it is still an object on your force, so you would not be able to have a resource.

The only characters that start the game with special objects that aren't considered part of your force are peanut-base Captain America, with The Shield and Scorpio, with the Zodiac Key, from the Captain America set. Teams with them can also use resources.

I would disagree that it is "pretty clear". I would never expect anyone to understand this without asking the first time they encountered a trait like this or without reading the old threads on this.
I will agree that your answer is correct though.

You are paying the point cost and assigning it to someone. I can't think of very many, if any at all, things that you pay points for but are not considered part of your force.

I can. Team members. I can pay 5 points to attach Black Queen to the Hellfire team base and not include a switch clix base for her in my sideline, thus making it so that no Black Queen is assigned to my force, but my team base is still complete.

Bummer. =[

While having a Team Member attached raises the Team Base point cost, those Team Members themselves are not worth any point value.

So, does that mean I didn't pay 5 points to attach Black Queen to the team base? Or assign her to it? I'm not sure I know the exact game term for describing this process of paying points for one game element (+5 to team base's cost) but instea am adding something that isn't that game element to my force (the team member's sculpt).

Ooh, that's a weird one: do I need the physical Black Queen sculpt then, if I wasn't planning to include her switch clix base in my sideline anyways? I think that may fall under the custom sculpt rules of a figure with missing or broken parts is legal and not a custom as long as the base is intact, so can I bring just the half-moon crescent to plug into the base and call it Black Queen?

The sculpt cannot be considered to be a deliberate attempt to misrepresent the capabilities of the played figure. The head judge is the final arbitrator of this decision

Sure, if it is truely broken then that 'half-moon' will match the rest of base.

Cool! Also, regarding the proper terminology, how do you describe the situation of the team base costing 5 more points but there not being an actual "thing" that raised its cost? Do we have an in-game term to describe paying points to attach team members? It would really help for when we need to discuss them, like we did in this thread.

Take for example the Utility Belt. You pay a cost for having slots open for the Utility Belt Items, but the cost is the Utility Belt, not the items being attached. The cost of the Utility belt is 8-14 points.

Similar to that is Team Dials, just like they have multiple starting lines with their individual cost. They also have cost per Team Member attached. But it is strictly the Team Dial cost, not the attached Team Member. If during the game you have a 300 point Team Dial and you use Solo Adventure ability, the Team Dial is still 300 points.

Relics, Feats, and ATAs that are assigned to a character are completely different and the rules tell us that those points are not added to the characters cost.

Thank you, that gives me the phrasing I was asking about! Team base + X number of team member slots activated (unlike the belt, these slots can't be changed each round, but that seems like the closest approximation).

Wait... you can change the belt items between games in a tournament?

Yes. During force construction you only activate the different item/suit slots on the Utility Belt. The actual suit and utility belt items are not attached until after forces are revealed at the beginning of the game.