Shutter button does nothing when pressed halfway, will not focus automatically. It doesn't make the beeping noise, no green box, nothing happens! I can press it all the way and take a picture, but t

When you depress the shutter button halfway, you are (in fact) activating the focus mechanism.
If in AF mode, you should hear the 'beep' when focus is achieved.
However, bear in mind that the XSi has three focussing modes: Quick mode, Live View, and Manual. There is a difference in how focussing is achieved between the Quick and Live View modes. (see page 102 onwards of the user manual "using AF to focus").
Also, the barrel of the lens will move during the focus operation.
If this does not happen, then there is a definate problem.
Having images in focus despite having no movement on the lens, simply means that the aperture settings are probably small enough to give sufficient depth of field to include your subject.

You cannot take a picture when your camera has not been focused yet. This is done by pointing your selected focus point on your subject, then click the shutter button halfway / half-pressed until you here a beep or see a dot on the lower left portion of your viewfinder. Once you have locked your focus, press the shutter button all the way down to take the shot.

TTTTTTTTTTTTTTTTTTTTTTTTITf correct focus cannot be obtained
When the subject to be focused is not on the AF target mark (not positioned
in the center of the composition), you can use the following steps to obtain the
correct focus on the subject. This operation is referred to as "focus lock".
Determine the position of the subject after focusing (Focus Lock)
1 Press the power switch.
• The camera turns on in the shooting
mode.
• The lens extends and the monitor turns
on.
• The orange lamp lights.
2 Turn the mode dial and select a
shooting mode.
"Shooting mode" (P.36)
3 Position the AF target mark on
the subject you want to focus on.
• When shooting a hard-to-focus
subject, point the camera at an object
about the same distance away as the
subject.
Press the shutter button halfway
until the green lamp lights.
• When the green lamp lights steadily,
the focus and the exposure are locked.
• When the green lamp blinks, the focus
and exposure are not locked. Release
your finger from the shutter button, reposition
your subject and press the
shutter button halfway again.
5 Keeping the shutter button
pressed halfway, recompose
your shot.
6 Press the shutter button fully.
Try the above, info from D-535, C-370 and X-450.
I can email you the complete manual if this works.
Regards
Phillip

If you check page 45 in the manual, you'll see that the orange light during the half press indicates that the flash is turned on. A single beep means that the camera is having difficulty getting a "focus lock" (usually when shooting in low light conditions). See page 100 of the manual for suggestions on how to improve the chances for getting a well-focused picture.

This depends on which Autofocus system that particular model uses.  Then manual should tell what is happening there- there is usually a more comprehensive manual on the supplied CD.
Failing that speak to the user support of you particular model

The shutter button features a two-stage action. Pressing Halfway Pressing halfway automatically sets the exposure, focus and white balance.
The indicators light or blink as follows.
Upper Indicator
" Green: Metering complete (two beeps sound)
" Orange: Flash will fire
" Blinking Orange: Camera shake warning / Insufficient exposure
Lower Indicator
" Yellow: Macro Mode / Manual Focus
" Blinking Yellow: Focusing difficulty* (single beep sounds)
* When the indicator blinks yellow, lock the focus before shooting.

What mode is the camera in? is it in "Auto" at the top there is a dial switch it to auto and press button halfway (shutter button) when you press the button halfway the camera will beep and display a square box on the LCD. If the camera beeps once and the box is orange it could not focus and you need to retry pressing halfway again. When it is in focus it will beep 2 times and it will show a GREEN box. This is the indication it could focus on the subject correctly. Its a machine and sometimes it takes it a bit for it to figure out what your wanting it to do. Its the nature of the unit

Try the FOCUS LOCK feature. This feature allows you to focus on a subject roughly the same distance away from you, then re-compose your picture without losing that focus. Open the lens barrier fully. Look through the viewfinder at the AF Target Mark (the 4 lines in the center of the viewfinder). Position the AF Target Mark on a subject roughly the same distance away from you. Press the shutter button halfway until the green lamp lights. While keeping the shutter button pressed halfway, re-compose your picture, then press the shutter fully.

Try the FOCUS LOCK feature. This feature allows you to focus on a subject roughly the same distance away from you, then re-compose your picture without losing that focus. Open the lens barrier fully. Look through the viewfinder at the AF Target Mark (the 4 lines in the center of the viewfinder). Position the AF Target Mark on a subject roughly the same distance away from you. Press the shutter button halfway until the green lamp lights. While keeping the shutter button pressed halfway, re-compose your picture, then press the shutter fully.

Try the FOCUS LOCK feature. This feature allows you to focus on a subject roughly the same distance away from you, then re-compose your picture without losing that focus. Open the lens barrier fully. Look through the viewfinder at the AF Target Mark (the 4 lines in the center of the viewfinder). Position the AF Target Mark on a subject roughly the same distance away from you. Press the shutter button halfway until the green lamplights. While keeping the shutter button pressed halfway, re-compose your picture, then press the shutter fully.