Photos first. (This is my preferred method but there are others you could use too):
Connect your tablet to your desktop PC using its USB charging lead. Your tablet should then appear as a new drive on your PC, just as if you'd plugged in a USB memory stick. (Note: With some tablets you might see an on-screen message, inviting you to use something like 'Data transfer mode'. If such a message appears, tap to accept). The tablet's drive will probably have several folders showing on it. Explore those to find where your photos are stored. (It might be in a folder labelled 'DCIM' or similar but there will only be a few folders to examine anyway). Then copy them to a suitable location on your hard drive (such as 'Pictures') in the usual way for handling files on Windows. [Instructions available on request but it's simply 'drag and drop' or 'copy & paste'].
Now Gmail:
Forget about deleting your email messages, per se. They're stored on a Google server, so deleting a message on one device will end up deleting it on any, and every, device trying to access that server. What you need to do is to remove your Google account from your old tablet. See here:
https://www.businessinsider.com/how-to-remove-google-account?r=US&IR=T
Once you've removed your Google account, the tablet will no longer automatically connect to that account. So it won't be able to access your Gmail messages (which is exactly what you want to achieve).