Can I start my IR with an FAA PPL? I mean, is it possible to add an EASA IR onto a non-EASA ICAO licence?
No, you will have to hold a valid EASA PPL.
FCL.610 IR - Prerequisites and crediting
Applicants for the IR shall:
(a) hold:
(1) at least a PPL in the appropriate aircraft category;....
GM1 FCL.005 Scope
(a) Whenever licences, ratings, approvals or certificates are mentioned in Part-FCL, these are meant to be valid licences, ratings, approvals or certificates issued in accordance with Part-FCL. In all other cases, these documents are specified.
FCL.610 does not specify an ICAO PPL and, therefore, means a valid EASA PPL. An ICAO licence validated in accordance with Annex III to Regulation 1178/2011 is not issued in accordance with Part-FCL (which is Annex I) and, therefore, does not count. In any case, even if you completed the course, the FAA would not include an EASA IR in the licence.