The laws of kilae begadim are taught in the 9nth chapter of masechet kilayim. First the Mishna says that the prohibition is only on wearing as the verse clearly says.
(Kilayim 9.2) הכרים והכסתות אין בהם משום כלאים, ובלבד שלא יהיה בשרו נוגע בהן
Pillows and blankets are included. The Rash explains, according to Gemara Yoma 69a, that from the verses clothes that cover the body are prohibited, and, as the Mishna says, even if they don't touch the body, e.g. if a kosher cloth isolates them from the skin.
For a pillow or a mattress: this last stringency is not mentioned because their prohibition is rabbinical. Chachamim have a doubt that perhaps a hair from the pillow will wrap around the body. When there is a separation, this Chashash is no more actual.
A saddle doesn't have the same rule as a pillow as written in Mishna 4, because it's harder. But there is an additional doubt that one may make from the saddle a patch for clothes and then the problem will be the same as for a pillow. One cannot wear a saddle on the shoulder, even to avoid getting dirty. If a hard fabric is placed on a padded mattress, it becomes equivalent to a pillow.
Towels are also prohibited d'rabanan because one may use them as gloves to protect hands. Rash in Mishna 3, following Rabbi Eliezer. Yerushalmi rules as Rabbi Eliezer.
Bath towels may be used to cover onself and are also prohibited rabbinically. Towels used at a hairdresser if there is an intent to use them as clothes are also prohibited. In Mishna Shabbat 5.3 they prohibit to keep in the hand ropes of wool and of linen knoted together because they can warm up the hand.
So in summary from your list
curtains -> as towels
carpets -> sometime a little padded and sometime have lengthy hairs
chairs, sofas and armchairs -> padded
handbags -> can warm up the hand
pillows -> meforash in Mishna
stuffed toys -> can warm up the hands
tablecloths -> as towels because they are sometimes used as towels
For the second part of the question, you need to check how this fabrics are made in the country they come from. In many cities in Israel there is a Shaatnez laboratory. You can ask where is that known problems. Nowadays Shaatnez is relatively rare.