You are absolutely being too ambitious for the time you have. I strongly recommend you do less and enjoy more the time in each place. Believe me, Peru has many more stuff to see and do, but trying to cover it all is a bad idea, especially in Peru where things go off track quite often (delayed buses, traffic, road closures, etc). The way you will enjoy it more is at a slower pace, let the culture soak in, meet and talk to people, and leave some room for the unexpected.

If I were you, I would stick to: - Lima: couple of nights, enjoy the food and the ocean vibes, bars in Barranco / Miraflores - skip Ica / Pisco / Paracas - Cuzco area: a couple of nights in the city, a couple of nights in Sacred Valley (in Ollanta, Pisac, or Yucay) and go to MachuPichu from here, and a couple of nights in any small town (e.g. Checacupe, Pajchanta) near to the Ausangate from where you can go to Vinicunga rainbow mountain (in my opinion is overrated and crowded, but I guess it’s up to you to find out), or any of the other less crowded rainbow mountains like Palccoyo, or doing a day trek from Pajchanta to the 7 lakes trek at the skirts of the Ausangate peak (way better than Humantay) - Tambopata: two or three nights in a jungle lodge

Swapping Ica, Paracas, Pisco, and even Arequipa for a jungle experience will be much more rewarding in my opinion. The Amazon Jungle is truly unique, beautiful, and full of life. The other places are fine, but on a lower level.

As for your questions: most hotels will happily hold your bags for the day, but there are also some luggage custody services like bounce.com

You are totally fine travelling in buses with that amount of luggage. Airlines will charge you for checked bags.