可能 can be placed by 可能是， 可能是 can work just like one single word. The following are all right to me: 我认为我(可能是)需要一个医生。 我认为我可能需要一个医生。 (我需要一个医生） 我想你可能太累(I prefer to put 并 in the end. ) 我想你(可能是)太累(了). (你太累了) Now look at some examples of another kind，in which 可能是 works as two seperate words， 我想这件事(可能)是他干的。 (这件事是他干的。) 我想这件事(可能)他干的。without 可能: 这件事他干的。(bad choice) So, you can always use 可能是 in both kinds of the sentences，but only use 可能 in the first kind，if you want to find out if 可能 is good enough，just remove 可能 from the sentence and see if it's still a reasonable sentence.

The first sentence is wrong. It should be 我认为我可能需要一个医生。 There is a rule from my experience 是 is needed if the word it is a non. 比如：他可是是一个医生。 If the word after 可能 is a adj , 是 should disappear. 比如： 他可能 累了。

可能是+noun (Indicate that something maybe is the another thing) 可能+others (可能是 in your examples are not so strange but "可能" is much more common)

It's not a strict rule! 可能=maybe,the meaning is not sure,so 是can't come after 可能.You just need say,我认为我可能需要一个医生 or 我可能生病了.