39-40 are so packed it feels absurd to say anything at all. Here are a few notes, but there is a lot more to say:
1. Regarding the sitting/standing debate - Assuming no nedarim are involved, I always recommend that people sit down, unless they feel their prolonged presence would be a burden for the choleh. Sitting tells the choleh he has your full attention, and you aren't racing out the door ASAP.
2. Why is there no gezeirah of "don't visit, lest you sit inappropriately?" I suspect it's because chazal don't want to create a gezeirah limiting bikur cholim. (Alternatively, perhaps we don't have a gezeirah because this is a milsa d'lo shchicha, but then why do we have a gezeirah not to sit, lest one remain?)
3. These laws re: neder pertain to private home visits, but should not apply to hospital visits, other than in an area where we pay people (such as social workers) to visit cholim.
4. The group of 7 is interestingly divisible into 3 complementary pairs, and Mashiach.
5. Note the angelic titling of Moshe as "Ben Amram." This is standard in Malach dialogue regarding Moshe, such as the gemara in Shabbos in which the malachim challenge Moshe's right to the Torah. We find in midrashim that "Ben Amram" is used as a pejorative, such as when Kalev gives his "bait and switch" attack on Moshe, and when Dasan and Aviram attack Moshe. Here, though, that doesn't seem to be the case. Perhaps it's because the malachim recognize Amram's great rank (as one of the four who never sinned). Alternatively, perhaps it's because Moshe was great in his youth, when he was known by his father's name - we have a similar explanation for the names of Ben Zoma and Ben Azai, in some rishonim.
6. Rav Yosef discuss the schar for bikkur cholim here. I haven't had a chance to do a CD-ROM search, but recall that Rav Yosef discuss schar mitzvah in other places as well - re: his own schar mitzvah as a blind man, and the question of the schar for an eino metzuveh v'oseh. Perhaps there are other places as well; I don't recall at the moment. See also his comment on Makkos 23b regarding HaShem's punishments and Chagigah 4b on dying early (I admit I looked at my own webshas notes for those last two).
7. A MUST see - The Ran at the top of 40a on praying for a person to die.
On 40, the issue of reward for bikkur cholim comes up. One of my Daffies here asked about the gemara at the end of Chullin, which says Schar mitzvah b'hai alma leca. I would respond with two points:
1. Specific to this issue - Here, the schar is explicitly stated in a pasuk;
2. In general - It seems to me that the nature of HaShem's schar/onesh is subject to machlokes in the gemara; views are expressed which are mutually exclusive. This isn't a big deal; as the Rambam noted in his comments to the end of Sanhedrin, we don't need to pasken on these issues.
40 is also very interesting for the tangent into the laws of Mikvah vs. Maayan. I highly recommend an article by Rabbi Howard Jachter on the differences between the two, and the issue of Zochlin.
You might take a look at Tosafos in Bava Basra 141a "l'didi", when you see Rav Chisda's note on the importance of a wife, on 41a. I'm not sure what to do with Tosafos's second answer, in relation to Rav Chisda's emphasis on marriage. There might be no connection, but it seems to me that there is one...
On 41 we find Rav Yosef's tragic story, which is particularly ironic given his role as "Sinai" in the classic "Sinai vs. Oker Harim" debate at the end of Berachos.
We also find a similar story involving Rebbe and R' Chiyya, and here the number 13 appears again. If you recall my comment on 13 and exaggeration, see my reference there to Bava Basra, but also see Rashi in Shabbos 119a on the ilisa d'dinri (towards the bottom of the page), and the Maharsha there. Of course, 13 here does seem to be a specific number - unless the point is that the launderer knew the lesser half.
Re: miracles of refuah, see the Ramban's approach to nisim, that the purpose of a nes nigleh (open miracle) is to make sure we notice the nes nistar (hidden miracle). We also talk about this regarding the nes of waking up in the morning.
Note also the Rosh on eating milk at the same table where someone eats dairy, and the difference between that and the neder case.
In general: See the Ran and the Rosh all through this daf; interesting conflicting views here, such as on the issue of the tamchui hachozeir l'baal habayis, and on the question of whether healing the animal is a mitzvah of hashavas aveidah.