Q: Power Macintosh 4400/200 (7220) – No video output

I assume that you read about the reasons for the 4400's design, with its use of "PC" type components (an IDE hard drive and an ATAPI optical drive).  Reduction of manufacturing cost was the driving force behind this.  Other Performa and Power Macs of that era that shipped with an IDE hard drive had a SCSI optical drive, until the beige G3s were introduced in late 1997.  If I wanted to restore a Performa/Power Mac of that vintage (with a very similar motherboard as the 4400), I'd opt for the 6400 or 6500 mini-tower.  The processors are the same: 603/180 MHz & 603e/200 MHz for the 6400s and the slightly faster 603e/(225, 250, 275, and 300) MHz for the 6500s.  I believe that the 4400 used the same L2 cache card as the 6400s/6500s, meaning that the cards were keyed (slotted) alike.  Because of this, the 4400 could use the same (third-party) G3 processor upgrade card, designed for the L2 cache clot.  If you can finally get that 4400 running, you might check ebay for a used/inexpensive one.  When they were still being manufactured by a number of companies (Sonnet and Newer Technology), they were an expensive upgrade.  The 4400 used the lower voltage EDO memory DIMMs.  The Apple (made by ATI) Rage 128 PCI graphics card that shipped with the B&W G3s is available here.  Ignore the part about the lack of a DVD decoder card.  The seller has another identical card with the decoder, but your 4400 doesn't have the processing horsepower required to handle it.  That decoder was designed to work in conjunction with the Apple DVD Player software 1.x, but the minimum processor requirement was supposed to be a 300 MHz G3.  It's a non-issue for the 603e/200 MHz processor in the 4400.  You mentioned that when you press the power actuator button on the front of the computer, it starts up and then immediately shuts down.  Does it do this, when you press the keyboard's power button in the upper right corner - assuming that your keyboard has one?  I'm wondering if the micro switch on the motherboard (that the actuator presses against) is damaged.  Internally, it contains a very thin metallic convex-shaped contact, that flexes inward against the rear contact when the power button's actuator presses against it.  The convex shape makes the contact naturally pop back out, when pressure is released.  If this internal contact breaks from the repeated flexing of years of usage, you could experience something strange like you've described, because it's supposed to be a momentary contact switch, unless pressed in long enough for a forced shutdown.  Otherwise, you may have to search online for the 4400's power supply pin-outs and check the leads with a voltmeter.

If you're interested, this Macworld article reports on performance testing of various G3 upgrade cards designed for the L2 Cache slot in those old Performa and Power Macs, including the 4400.

Hearing one beep on startup indicates that no memory is installed.  If you removed the RAM to inspect for corrosion, did you snap it back in so it's fully-seated in the slot?  If the RAM is completely incompatible, it may not be recognized as being installed in the slot and generate the same single beep.

Just for clarification, when you referred to the sound as a "beep," were you referring to the normal startup tone?