Schema review for system to advertise cars
Initially I have thought to only have reference to "car_model" in "posts" table. Cause technically I could access the car make via the "car_model" table. But then decided that for search purposes it would be easier to have both keys in this table, correct decision?


No,
this adds redundancy to your database and this is something you generally do not want to have at all. It also does not fit with the 3rd Normal Form.

To read about Normal Forms look [here][1]


  Another point, is with properties.


It is no problem and very typical to have database querys across multiple tables. (This is basically why we use databases and have multiple table and not smash everything together) Do not worry about it and please do NOT add extra redundancy with such an extra text field.


  any other suggestions given the specification?


Yes, there is no need for the idproperties_post_rel column in your properties_post_rel table. You should make the both foreign keys the primary key so you can not accidentally set a property on one offer twice. 
Also the relationship is well enough defined by the unique combination of these FKs so no need to waste space for an extra synthetic key.