1. You deal with the last-delivering carrier. That is LH. LH is responsible for settling up with you no matter who actually caused the bag to go missing.
2. Most bags show up in 1-2 days. LH will generally cover immediate needs such as toiletries, but I would not spend anything without prior authorization from LH if you are looking at more. This is especially true given generous carry-on allowances which should allow one to carry on the first few days clothing.
If your bag is not located in 20 days, it is considered lost and you will be entitled to collect fair market value up to a maximum of approx. $1,600. Anything which LH pays out in advance will be deducted from that $1,600 (if the contents have a FMV that high).