三个星期前我开始每天早上跑一个步。

跑个步 is perfectly fine in spoken language.
If you want your writing style to be more written, you can just say 跑步

我还不可能跑得很好。

我还不可能跑得很好。= It's impossible for me to run very well right now.
我还不能跑得很好。= I can't run very well right now.

我跑步的时候需要散步花很多时间散步，我不会完全把所有的时间用于跑步。

The meaning of the revised sentence is "When I run, I spend a lot of time walking. I don't spend all my time running."


花很多时间散步 = Spend a lot of time walking
花（Spend） 很多时间（a lot of time） 散步（walking）

“walking a lot of time” may be ok in English,
but in Chinese I don't say 散步很多时间
I say 散步很久

so you can also say 我跑步的时候需要*散步很久*，我不会把所有的时间用于跑步。

完全的时间 sounds a little strange
you can use 所有的时间(All of the time)

开始每天跑步以后，我发现跑步不需要没有那么多痛苦。

The meaning of the revised sentence is "After I started running every day, I found that running was not so painful."

you can also say
“开始每天跑步以后，我发现跑步*不怎么痛苦*。”
"不怎么痛苦" has a similar meaning to "没有那么痛苦".

If you have to use the word "不需要", you can say:
1. 开始每天跑步以后，我发现跑步不需要承受那么多痛苦。（After I started running every day, I found that running didn't need to bear so much pain.）
2. 开始每天跑步以后，我发现跑步不需要付出那么多辛苦。（After I started running every day, I found that running didn't need to pay so much effort.）

我跑步的时候需要散步很多时间，我不会完全的时间跑时，大部分时间在散步。

开始每天跑步以后，我发现跑步不需要那么多怎么痛苦。

现在我连喜欢它一点儿竟然有点喜欢它。