The 5th gen "iPod video" was released in late 2005, so it's not quite 10 years old.
The "new" battery may be defective. The inventory of vendors who sell batteries for old iPods may be getting old, and lithium batteries can become unable to hold a charge if they are not discharged and re-charged regularly.
If you have not already, you should obviously open it up again to confirm the connection point of the logic board to the battery is not loose. Also, the battery part seems to have a long delicate ribbon cable, so visually check that cable for any damage.
Does your iPod appear in iTunes or on your computer as external storage, if you connect it now with no battery inside. I know older iPods (4th gen and earlier) function normally as an iPod without a battery connected, as long as it is connected to power with the docking cable. Not sure about the 5th gen iPod.
If it you can get it recognized by the computer, you can off-load the songs. iTunes syncing is generally in one direction only, from the computer's iTunes library to iPod. So you cannot use iTunes to transfer song files from iPod to computer (except for songs purchased from the iTunes Store). However, there are third-party methods and utilities that can transfer from iPod to computer. If you do an Internet search on something like "ipod transfer," you should get some links. (The CNET article seems to be popular.)
I guess you do not have the original photos? When photos are synced by iTunes to the iPod, iTunes optimizes the photos to fit the iPod's small screen and take up less space on the iPod, and the photos are stored in a single database file (instead of separate image files). The only easy way to recover photos from an iPod is IF (when you synced the photos), you had a setting enabled on the iPod's Photos screen iTunes (where you set up syncing for photos). The setting is to "include full-resolution photos when syncing" (that may not be the exact wording). If that setting was enabled, iTunes puts a full-resolution copy of the image files on the iPod, in a folder called Photos, which you can access (using Windows Explorer or Finder on a Mac) by turning ON the Enable disk use setting.
Otherwise, the photos on the iPod are low-res versions of the original photos, and they are not directly accessible on the iPod. There may be third-party utilities that can access the photos on the iPod, and save them off on your computer's drive.