Here is a tip for training your team's gym in order to make room for your own Pokemon. It will be added to the OP.
What is the easiest method of adding my own Pokemon into a Gym that already has some of my team's Pokemon in it?
As long as the gym isn't capped at Level 10 then you will have the opportunity to add your own Pokemon in by training. Hopefully you have a Pokemon that is able to at least defeat the first Pokemon at the Gym because that is essential. Battle the Pokemon defeat it and as soon as you get to the Pokemon that you can't defeat either due to the CP of it being too high or you are running low on HP use the 'Run Away' button, bottom right of your screen that looks like a Pedestrian symbol as quickly as possible. Heal up your Pokemon and rinse and repeat until the Level of the gym gets high enough where you can add your own Pokemon into the gym. Note that the Gym will gain more Prestige/HP if your Pokemon that you are training with is lower or almost equivalent to the CP of the Pokemon you are defeating. The higher the Gym's Level is the more Prestige/HP you will need to add in order to make room for your Pokemon. Hopefully the Pokemon that you choose to leave at the Gym isn't lower than what you've been defeating because once rival Gym members try to take it over it'll be the first to go.
What coordinated groups do is put a lower CP Pokemon as the first Pokemon and they continuously defeat that one Pokemon to raise the Level/HP of the gym to very high levels before adding their Pokemon. Usually they'll get it to Level 7 or so, since groups usually consist of just a bunch of friends who can fit into one or two cars.
Just hope that you are the only one in the area while you're training because other team members can and probably will take the open slot if it becomes available. And if you notice the Prestige/HP of your Gym going down you may as well stop because other teams are battling at the gym and it is much easier to take the Prestige/HP of the Gym down than to build it up.
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I'm unsure if this method can also work for taking down rival Gyms since I don't know if running away while you are battling another gym will add Prestige/HP to the gym or not because it may count as a Win for them.