Unfortunately for that kind of budget your options are going to be very limited. First of all, receivers (even high end ones) typically have only one or two HDMI outputs. What you need to be looking into are matrix switches. If you only want to do HDMI then there are some cheap options out there for some basic 4x4 switches (4 inputs/4 outputs) in the $200 range. You can also get relatively cheap HDMI-CAT5/6 baluns in the $50-100/pair range be be aware that they can be flaky. I've seen cheap ones have problems with frequently losing sync due to interference from appliances, fans, lights, etc. and other times they work fine so it's a real crapshoot.

Then if you want to introduce distributed audio with in-wall/in-ceiling speakers you now need an audio matrix switch with a multi-channel amplifier. However, even a cheap one of those approaches your total budget. Plus you have the remote as well. For that I would only consider RF since IR distribution will likely cost you the same or more in the end and be a lot less flexible. However, here again you're into $500-$1000 or at least a few hundred for a "high end" Harmony remote.

Bottom line is I don't think it's likely you can put something together with those sorts of capabilities with that sort of budget. An entry level system would typically be much more in the $3,000-$5,000 range. You could maybe cobble something together that falls more into the $1,000-$2,000 range but be prepared for a lot of tinkering and pulling your hair out.