Function of particle を is not only to indicate object for transitive verb but also to indicate location where moving action goes through for the verbs which have meaning of moving.
散歩【さんぽ】する is intransitive verb so that 公園【こうえん】を is apparently not object for さんぽする.
Below verbs belong to this category.
歩【ある】く 行【い】く 走【はし】る 進【すす】む 渡【わた】る 通【とお】る 越【こ】える 過【す】ぎる 泳【およ】ぐ 潜【くぐ】る 飛【と】ぶ 転【ころ】がる 横切【よこぎ】る 曲【ま】がる 登【のぼ】る 降【お】りる 上【あ】がる 下【くだ】る 下【お】りる 下【さ】がる 動【うご】く 移動【いどう】する・・・
散歩【さんぽ】する is certainly one of them. Principally above verbs are intransitive so that location+を should not be considered as object.
So, your first question of こうえんをさんぽします is correct sentence with indicating こうえんを as location for walking around.
As far as こうえんにさんぽします is concerned, because に for location indicates target point or point where action gives effect, に doesn't match with さんぽ. In this case, particle で for location where action is done is suitable.
About second question, I would like to introduce you する-verb system at first.
Some nouns can be directly attached verb する without を so that it becomes one word verb like さんぽ＋する＝さんぽする. It is authentic verb vocabulary like たべる or あるく. You don't need to suppose it as omitted/abbreviated phrase without を.
But please be careful that not all the nouns can be treated like this. Only nouns contain meaning of action and have been accepted by natives are vital points.
Now, you have two options to express it, さんぽする and さんぽをする. You are free to use whichever you like.
If you want to mention the place for さんぽ, you can use こうえんを or こうえんで, connotation of former is walking around on the place and latter to walk around inside of.
If you would choose こうえんを, you had better avoid さんぽをする. Double use of を will probably make listener annoyed.