JotForm is a free online form builder which helps you create online forms without writing a single line of code. No sign-up required.
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Yes you can add custom logo in your free account. Apart from the limits on submissions and upload storage space, there is no major difference between any plans we have.
You can add logo to your form by adding "Image" question of your form.
When you add image field, you will be presented with a wizard where you can upload or provide url of your logo. You just need to follow the steps.
On your second question:
In free accounts you can receive 100 non secure, 10 secure/SSL and 10 payment submissions. If any of these limits are reached, all your forms will be disabled. Your form users will not be able to submit your form. To activate your form again, you will have to upgrade your account or wait for the first day of next month when the submission counters are reset.
In free account you have an allocated upload storage space of 100 MB. Again, if you reach this limit your forms will be disabled. Unlike submission counters, the upload storage space is not reset every month. You need to delete your submission records to free up your upload storage space. Deleting your form's submission records will also delete the attached files.
Hope this helps.