I am conducting a systematic review, and I decided to not limit the design of included studies to a single type. Can I use the different risk of bias assessment for each study due to the different designs?
Yes, this is fine, as long as you can argue well the inclusion of more than one study design type.
The most difficult thing is then on how you could compare the different study ROB tools when study designs are different. It really depends on your question of interest and how you could really compare ROB (ex for observational vs interventional studies) as well as the specific area you are working. For purely descriptive purpose, I think it is relevant but it would be difficult to compare different studies with 2 different ROB tools.
Yes you can use different critical appraisal tools. This will depend of the type of study you are going to include in your review. Is it is important to see if this type of study will be relevant to answer your research question and how you are going to analyse these different data from each study.