You're probably talking about a
DOCSIS modem, and apparently the one provided by Suddenlink
* is not VOIP capable, hence the requirement of a separate phone (VOIP) modem. The lack of a single cable/VOIP modem to handle twice your current speed is somewhat strange, but that may simply be a cost/availability issue, particularly if you get the modems for 'free'. This is something to ask them about (but see below).
That said, a 10Mbps hookup is plenty fast for the needs you describe, assuming that's what you actually get. You could test that with web sites like
this one. So,
1. the 10>20Mbps upgrade may not be all that earth shattering, but should be noticeable with bandwidth-critical data like (Hi-res) streaming video, and generally with all data served by sufficiently fast servers.
2. Video chat, like every conversation, is a two (or more)-way street. It sounds that you're experiencing less than optimal performance here. Note that the speed on your side won't affect the speed on your conversation partner's end, which may in fact be the limiting factor. And let's not forget that the (speed of the) computers involved plays a role too.
3. DOCSIS modems may suffer from the same issues as any other modem, but there is a range of quality in the available devices (see below). Without specifics it's hard to say where your ISP's equipment falls on this scale. It's likely adequate, but perhaps not great.
4. A DOCSIS modem should be compatible with most routers like Airport. You'd simply connect the two with an ethernet patch cable like you likely do right now.
*) In cases like yours it may be cheaper in the medium to long term to own your own modem, especially if your ISP charges a monthly equipment fee for theirs, or requires a minimum subscription duration (1-2 years) to offset its cost. If this applies to you I can recommend a Motorola DOCSIS 3 SURFboard digital voice modem like the SB6120 or SB6121 (up to 100Mbps, available for around $85). If Suddenlink allows you to use your own equipment and waives their fee, you could earn it back pretty quickly.