It would not be considered to be wrong, specially if they know that you are still learning the language. I would highly encourage you to look for the kanji if you are writing and you do not remember one of them. If you write a letter and write only some kanji and not others, it can be considered as laziness from your part. Japanese are very detailed in everything they do and not writing properly their language can be considered as lack of respect. If you are in a conversation and you forget a kanji, you can always ask and it will be considered as a great effort on your part, people will see that you are doing your best to learn and treat them and their language repectfully

Nothing grammatically wrong. You can find lots of those mixed-words on references for early elementally graders. It's just the matter of readability. Referring to a dictionary every time is great and pretty diligent we see, but if it annoys you a lot and if it ties you up, I recommend you to write easy. Just imagine the spellings in English. Anyways seeing the references I mentioned will help you figure out your question for sure :)

It doesn't present any problems. Japanese children grow up using kanji little by little. In my opinion, using half kanji or writing all in hiragana is much better than using wrong kanji, for example, 映語、英後 、etc. However I always admire people who study Japanese, even though writing wrong kanji is not bad if they are corrected and remember the correct kanji. By the way, there are cases that writing in hiragana looks soft and lovely. For example, I like both of writing 桜 and さくら、頑張って and がんばって.

It is a matter of notation, not grammar. Of course, your examples are all grammatically correct. Notations like that are called 交ぜ書き (mazegaki), and it is true that some native speakers do not really like them. However, Mario is definitely right. It would never matter if the reader knows that you are a learner. That being said, I would suggest you try to avoid that way as much as you can.