Is there a way to make cell calls over the internet? Does it have to be WI-FI or can you program some kind of network integration infrastructure system?

sure it is possible; it depends on what kind of equipment and servers do you have or how much would you like to invest.
Another issue to be considered is that due to regulations, it is not a legal operation everywhere.

WI-FI uses 802.11b and 802.11g type radio devices simply to extend the wired (ethernet) internet network, so regarding feasibility for making make phone calls whether the internet network involves is Wi-Fi or not does not matter.

If you want to make phone calls, starting over the internet you need a VoIP phone (or equivalent software in a PC, e.g. Skype).  We have some hand held WiFi VoIP phones and they work fine in the local wireless LAN surrounding a VSAT terminal.   Calls may be made directly with other local handsets and calls to the PSTN via the satellite and the teleport hub.

A VoIP call goes immediately into the internet (via WiFi or ethernet).  The call then goes either directly to the IP address of another VoIP phone that is also connected to the internet or the call goes to a SIP server gateway where the G729a voice coding is changed (needs digital signal processing DSP hardware) to G711 and the call transitions physically to the PSTN, via which it may be terminated at a PSTN phone.  You need a gateway device (e.g Cisco router with CME and DSP farm) physically linking the ethernet internet to a number of PSTN lines (e.g. T1, E1 or ISDN cable).

Regarding feasibility of making calls from and via the internet to mobile cell phones, this will depend on the configuration of your gateway.  Does your gateway permit numbers to pass to the PSTN which will end up on mobile phones ?

In the opposite direction, from a mobile cell phone to an internet VoIP phone, it is possible, provided the gateway is set up suitably.  From the mobile phone you need to dial a PSTN number which is assigned to one of the lines on your gateway and the gateway needs to be programmed so that such calls get routed to a particular VoIP phone according to its IP address.  The VoIP phone has an IP address and also a PSTN phone number assigned to it for incoming calls.

Don't underestimate the cost, complexity and time needed getting it all to work properly.  It may never work perfectly.

If you buy the interface service from a third party make sure you understand how the billing system works and try to find out if the third party actually knows how to correctly configure their equipment to match your network.   Ask for a demonstration that matches your network configuration.   Ask how quickly they can respond if you change your public IP address?   If you buy equipment check that you are clever enough to operate it.  How much will technical support cost ?   Will the equipment tolerate unplanned power failures ?

I am only a beginner at learning how VoIP works so more comments here by others is requested please.  Also corrections if I have got it wrong.

Best regards, Eric.  

What kind of equipment and servers are needed?

Regarding the end user VoIP phones I can recommend these:

VoIP wireless phone Hitachi IP3000 brochure (197k bytes pdf file)

VoIP wireless phone Hitachi IP3000 User manual (1.2 Mbytes pdf file)

Regarding hub VoIP equipment, router, SIP server and gateway hardware I too am very interested to hear of recommended devices from anyone who has had good experiences.

Best regards, Eric.