پیشنهاد دادن

It is neutral. It works with friends and in the office. 'Pishnahād kardan' is the more formal version.

Yes, 'pishnahād-e ezdevāj dādan' is the standard term for proposing marriage.

'Pishnahād' is a suggestion (an idea), while 'tosiye' is a recommendation (often based on advice or expertise).

If you are following it with a verb (e.g., 'I suggest that we go'), yes, use 'ke'. If you are suggesting a noun (e.g., 'I suggest this book'), you don't need it.

You say 'Pishnahādet chie?' (informal) or 'Pishnahād-e shomā chist?' (formal).

Yes, 'pishnahād dādam' means 'I suggested.'

Constantly. You will hear it regarding peace proposals, budget plans, and international deals.

There isn't a single word, but 'rad kardan-e pishnahād' (rejecting a suggestion) is the functional opposite.

Yes, it is very common and polite, especially if you add 'be nazar-e man' (in my opinion).

Only in the sense of 'offering an idea.' For offering a gift or a price, other words like 'arāye dādan' or 'pishkesh kardan' are used.

It's pronounced 'peesh-na-haad'. The 'aa' is long like in 'father'.

Yes, 'pishnahād kardan' is perfectly correct and slightly more formal/literary.