I would say (1) remember that anything she requests is temporary because this is a visit and (2) try to remember that if she seems tense, it could be culture shock and insecurity as much as anything, so try to be as accommodating as possible.
-Most Russians do not let babies cry. However-
-They do feed children a lot of sweets.
-They let small children (toddlers) drink tea with sugar.
-They swaddle. If baby does not cry, let her swaddle baby if she wants. If baby does cry, explain that this is from your side of the family, but that baby will be fine.
-If, God forbid, baby should sneeze or cough during the visit, and she wants to medicate baby, this could be quite worrying because she will probably really want to help (as the grandmother, naturally! who can blame her) but also may have different ideas about what is appropriate. Make sure she understands that here, we never give baby anything
without the doctor's express consent. Yes you may end up calling the doctor for stupid things but it's better than arguing with MIL about whether baby needs to have some bizarre Slavic home remedy inflicted on her for clearing her throat.
I don't think they over-bundle their children. I have never heard of a Russian child dying of heat stroke in September. I would say that many Russians have a slightly more old-fashioned take on dress (keep your hat on) but this is from a people that have kids walking to school in -30 C as well. If she wants you to put a light cotton hat on your baby, do it. Really, it is not going to hurt the baby. Same with socks. If she really goes to far, tell her, "In this climate, the cool doesn't affect the baby in the same way- it is a light, dry breeze."
Emphasize the different climate. When I was in Russia, they kept telling us, "Yes, it was this cold in America but this is a different climate." So I believe the "climate" argument should hold some weight. If she personally feels chilly, tell her her body is still adjusting to the climate, baby is adjusted, thank you, but offer MIL a sweater.
Russians wear "house clothes" indoors, which saves a lot of washing. You may be expected to provide slippers, but otherwise do not be surprised if your MIL appears to wear pyjamas all day. They are not technically pyjamas, they are house clothes.
Remember that as a grandmother, she has been dying to see this baby and you get to see the baby every day. Yes you are the mom, but in Russia, babushki hold a special place in the child's heart. Let her have her time. If you miss the baby, tell her so, but try to remember that she may want to just soak up every bit of grandbaby she has in a limited amount of time, and respect that. She will be so far away when she is back in Russia and she won't have her own son there, either. Try to understand that if she seems posessive of baby. It is because it is for so little time.
"I think smiling is seen as a weakness though and they seem brusque on the outside."
No. It is not a weakness, they just don't go around smiling for no reason. I think they are more thoughtful and introspective in temperament and enjoy introspection and a little moroseness now and then. They do not express happiness through smiling or friendship through words. It is something deeper than that. They like it when we smile at them, provided they have reason to believe it is genuine. At least, people who are not arrogant teenagers generally do.
I found that Russians usually have soup for dinner, not a cold dish! Though, breakfast is sometimes cold.
Ask your MIL to prepare and teach you as many dishes as possible. She will feel needed and appreciated. Even if you don't plan to make it again, who will it hurt and it will give her such joy to think that her son could get these dishes in the U.S.
Your MIL might expect you to be as coy as Russian women are to their husbands. She might find you very demanding and forward, as a woman. Tell her it is your culture but if she wants to give you tips, she is welcome. Again, remember that this is a visit so if she spends two weeks being happy by "fixing" her daughter in law, it's her vacation, not yours, it will be over soon. Smile and relax- she is just doing what she can.
Russians love to take walks unless they are ill.
Russians believe bananas make you happy (this is actually somewhat scientific). If MIL seems sad, buy bananas for her to show her you care about her well-being.
When a Russian feels neglected, sad, lonely, stressed or whatever, he or she may develop a psychosomatic illness like the flu, and expect to be pampered. Blood pressure is also something that can suffer. If your MIL seems sad, offer vitamins as a precaution. If you personally take a certain brand, and can afford the same brand for her, try to help her feel cared for by offering a similar vitamin and sending some home with her. Maybe this could prevent a cold or flu if she does start to feel homesick or lonely.
Russians like to have long chats over tea, talking about their problems. Even if you don't speak Russian, invite her to chat or teach you Russian over tea while you breastfeed or whatever.
My MIL is not Russian but I lived with a Russian babushka and I fell in love. They are very opinionated and demanding but they have earned it. You just have to kind of enjoy being the child for awhile and go with it. Of course you have your limits and it's your house but the more you can let babushkaness flood your home the more you will enjoy it. They might look cold and brusque but that is just the outside that holds in so much love.
Good luck! Where is MIL from, by the way, and what does she do for a living?