yīshēng shuō tā de shēntǐ bú tài hǎo, yào zài yīyuàn xiūxi.

By itself, 他 is ambiguous; it could be the patient or another male the doctor is talking about. The second clause 要在医院休息 shares the same (implicit) subject as that 他. To disambiguate:

• If it’s the doctor himself: 医生说自己的身体不太好，要在医院休息。
• If it’s a specific patient: 医生说小王的身体不太好，要在医院休息。
• If it’s you: 医生说我的身体不太好，要在医院休息。

Here 要 expresses requirement/recommendation: “needs to/must.” Alternatives:

• 需要: need (objective necessity). 含义接近: 需要在医院休息.
• 得/必须: must (stronger obligation). 得在医院休息 / 必须在医院休息.
• 应该: should (advisory). 应该在医院休息.
• 会: will/likely to (not obligation). 会在医院休息 = “will rest in the hospital.”

With body parts and close relations, spoken Chinese often drops 的: 他身体不太好 (colloquial). 他的身体不太好 is equally correct and a bit more explicit/formal. Note:

• 的 marks possession/modification (e.g., 我的书).
• 得 marks complements of degree (e.g., 说得很好).
• 地 makes adverbs (e.g., 认真地学习).

Not exactly.

• 在医院休息 = rest at the hospital (location emphasized; could be short-term).
• 住院 = be admitted/hospitalized. Doctors often say 要住院 or 需要住院观察. You can also say 要在医院休息几天/住院几天.

Put the duration after the verb phrase:

• 要在医院休息三天/两周。 With 住院:
• 要住院一周。 For completed time: 他在医院休息了三天。

Chinese doesn’t need a “that.” 说 + clause is standard. The comma links the two parts of what the doctor said. You can also write:

• 医生说：“他的身体不太好，要在医院休息。” If you want a logical connector: 医生说他的身体不太好，所以要在医院休息。

Not required. Chinese often leaves tense to context.

• 医生说…… neutral; context supplies time.
• 医生说了…… emphasizes the completed act of saying.
• 医生说过…… = “has said (before).” Add time words for clarity: 昨天医生说……

休息 is intransitive; don’t attach an object like 身体. Natural patterns:

• 休息一下 / 休息一会儿 / 好好休息
• 在家/在医院休息
• With duration: 休息三天