Yes as to France and Britain going to war.
No. This cannot be assumed. Hitler's betrayal of the recent Munich accords was what persuaded the British that he could not be trusted about anything and that seeking further compromise with him about German claims was useless.
If Nazi Germany leaves the Czechoslovakia rump with nominal independence (there was no doubt that geopolitical and geoeconomical realities would place it in the German Sphere of influence sooner rather than later, but economic and political penetration of Germany in Central and Eastern Europe was not a problem for Britain, outright military conquest was), this shall be seen as good proof that the Munich approach is productive and and the same method be applied to any claims that Germany will make on Poland. Of course, Bohemia-Moravia was rather more economically precious than any other possible conquest in Central and Eastern Europe, except Ukraine and Romanian oilfields, and historically it was an integral pat of the old Reich, so it was somewhat hard to give up its integral possession for Nazi Germany expansionism.
The Polish will be given no blanket garantees nor military alliance by the UK. Britain and France shall seek a compromise between stated German claims and what they deem as "reasonable", and if any can be found, they shall give Poland the choice of accepting it or facing Germany alone.
Whether Poland would accept it or not would be anyone's guess.
As to how it effects things, Chechoslovakia probably follows the route of supplying the Nazis to help prevent invasion.
You would have to ask why, given Hitlers mentality he would not invade though.
Well, this is the crux of the matter, wouldn't it ? Why Hitler would give up conquest of Bohemia and Moravia. After Munich, he regretted the deal had prevented him of the opportunity to ride in Prague as a conqueror. Theoretically, he could have easily used economic and political pressure (the Slovaks were in Germany's pocket, and with them, Czechia was completely encircled) to make them a fascist satellite much like Bulgaria, Hungary, or Romania. True, Germany would have not able to exploit Czech industries, natural resources, and weapon stashes (their tanks were good) as efficiently as with occupation, nor use their gold reserves to stave off German debts, but "soft" dominance would have caused no outrage in London.
Therefore, either Hitler has a flash of insight, and realizes occupation of Prague would push the UK government against the wall, or he is no longer in charge. of course, this also matters about the true aims of Nazi Germany in Poland. Historically, they mainly sought to make Poland a colony. Now, if by the same change of strategy that causes the PoD, they would instead seek to make Poland a German satellite, and only annex their long-standing national claims (Danzig, the Corridor, Upper Silesia), it is highly probable that without the betrayal of Prague, the UK will be quite amenable to concede that in Munich II. And France shall always follow the lead of London, no matter how reluctantly.
If Poland accepts, the road is paved for a clash of titans between Germany and the USSR in a couple of years, with Britain and France spectators cheering for mutual exaustion or a German compromise victory. Or possibly, is Stalin makes something dumb that paints the URSS as the true aggressor, say an invasion of Finland or Romania, an anti-Soviet alliance between Britain and Germany. Again a WWII, but with URSS as the rogue state. Italy would enthusiastically join the anti-communist crusade, France might as well or stay neutral. What would Japan do is a wild guess, they might just as likely attack in Siberia or go South. The USA shall stay neutral unless Pearl Harbor occurs.
If Poland refuses a Munich deal, well, Paris and London shall wash their hands clean of the stubborn Poles, and Warshaw will be crushed in a quick campaign. The URSS will invade Eastern Poland very soon after the start of the Polish-German war, with or without a previous deal with Germany, to protect their claims.