What you’re describing sounds very much like what I’ve experienced a lot.
For your first question: as far as the body is concerned, stress is stress. If it feels like stress, then it is stress, which means respecting that and maybe backing off the training, doing extra self-care, sleeping more, eating more, etc. So, yes, in those weeks, the best thing to do is to really listen to your body. Maybe do yoga instead of strength training, add in an extra rest day, do a recovery run (zone 1), go for a swim or easy bike ride rather than running/climbing/etc. In those weeks, I give myself permission to actually not do anything beyond easy walks, if I don’t feel up to it. A good 1 hr yoga practice with some arm balances can be a good way to get in some low-key strength work. One week every now and then is not going to derail your training over the long term. It is important though the following week not to ramp up a lot right away — maybe repeat the week you did before the “down” week, or just add 5% rather than 10%.
These are some good articles on strategies for recovery:
If you start to have these difficult weeks more regularly, you could cycle your training so that the recovery week coincides with that time of the month.
As for strength training in the second half of the cycle, by all means, if you feel up to it, do it! Yes, increasing protein intake can help. There is also increased need for hydration during that phase (the luteal phase) and increased need for carbohydrate for recovery. Think: “recover with carbohydrate!”
As far as nutrition in general, the body can’t store protein, so the most important thing is to make sure you’re eating BOTH protein and carbohydrates throughout the day, breakfast included.
UA dietitian Rebecca Dent recommends “Uphill Athletes consume 1.4–1.6 g/kg/BW), and for some a higher protein intake of 2 g/kg/BW may be necessary. Protein intake should be spread evenly throughout the day at 4-hour intervals as meals (~0.3 g/kg/BW) and snacks. This protein should come from good-quality protein sources that provide the essential amino acids required for muscle mass (e.g., animal proteins such as meats, fish, dairy, and eggs and complete plant proteins such as tofu, Quorn, hemp, and quinoa).” (from this article: https://uphillathlete.com/nutrition-injury-recovery/ )
The overall guideline is: Listen to your body and respect what it’s saying. This is a lifelong process, not a sprint.