The FAA Airmen Certification Branch is usually pretty quick about requesting foreign licence records. Delays are almost entirely caused by the foreign authority. Once again, the verification can be requested through IACRA, which saves you filling in paper/pdf forms. You will need to register on IACRA to obtain an FAA Tracking Number (FTN), which is needed to book a knowledge test.
Unless you're working for the US DoD, you'll need to do the FAA knowledge test within the United States. There is no instructor endorsement needed for the first attempt at the ICP, IEP, and IFP knowledge tests. You can verify this by checking page 11 in the FAA
Airman Knowledge Testing Matrix. The knowledge test can be booked through PSI at
https://faa.psiexams.com/faa/login
An IPC is needed to exercise instrument privileges and the check must be passed
before applying for a standard US IR under the EU-US conversion agreement. You may apply for a restricted US instrument rating, pursuant to 14 CFR 61.75, without passing an IPC; however, an IPC is needed to use the rating.
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after 10h acclimatisation flights with TSA"
No, the TIP-L does not prescribe a number of hours for the acclimatization (training). That is left to the discretion of the US flight instructor. You are exempt from completing this acclimatization (training) if you have 10 hours of instrument flight time as PIC, logged after obtaining the EU Part-FCL IR. So, if you don't have those hours, go find a buddy or mentor through NAFI/SAFE/EAA/AOPA or whatever and either do simulated instrument flying or use a restricted instrument rating to fly in actual IMC. NB: if you're flying in actual IMC, you'll need an IPC to first establish instrument currency.
Once you've received the verification letter, you can apply through IACRA for a standard US private pilot certificate and/or standard US instrument rating, to be issued iaw the EU-US conversion agreement.