When making a contact that is a spirit, is the connection rating the force? Or how is the force calculated?

You could make the connection rating the force but there isn't any hard and fast rule on it either way. You could make it up or use connection or roll a die or whatever. Up to you as the GM really. Though I wouldn't make a force 1 spirit connection 6 for example so a little common sense should come into play but otherwise, just go with whatever seems right.

So...
Could have
contact 3/4
airwave spirit force 4 Nickname "Bogart"
spirit marker 3 (the rules suggest this but do not give cost for this market besides role-playing costs) ?
Char has an old trivid set predating the wireless matrix. That is powered up but not connected to anything for the spirit to communicate through.
Wears an emblem of the spirit as a shoulder patch and had a tattoo on the shoulder.

Is this too much? For 7 karma for a starting char?

7 karma is nearly a third of your starting karma (without negative qualities) so no I don't think that's too much. Force 4 isn't really that powerful in the realm of spirits anyway. I'd allow it so long as the GM makes sure to play whatever negatives come with the spirit marker.

This is covered loosely in Forbidden Arcana, but does not reference Force. I think Force is purely determined by the GM. You may have a connection 6 spirit with F1 (unlikely) or F8 or more. I would use connection as a rough guide; since the interactions "feed" the spirit, I would expect a spirit with a lot of contacts to be "bigger."